These notes cover PMATH 450 (Spring 2018), taught by Prof. Stephen New. The course develops the Lebesgue theory of integration on the real line, culminating in the \(L^p\) spaces, Hilbert space theory, and Fourier analysis. The primary source is Prof. New’s own typed lecture notes; Chapter 5 on Fourier series draws from Calvin Kent’s student notes, since Prof. New deferred that material to the textbook (Real Analysis by Bruckner, Bruckner, and Thomson, §§15.1–15.5, 15.11).

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These notes are primarily based on Prof. Stephen New’s lectures (Spring 2018). Additional definitions, theorems, examples, and explanations have been incorporated from Felix Zhou’s course notes (Prof. Katherine Hare, Spring 2020).

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Chapter 0: Foundations (Set Theory and Order)

Before developing measure theory, it is worth collecting the foundational tools from set theory that appear throughout the course. The Axiom of Choice and its equivalents — Zorn’s Lemma and the Well-Ordering Principle — are invoked at several key points: in the construction of a Vitali non-measurable set, in the existence of a Hilbert basis for every inner product space, and in the construction of a basis for every vector space.

Partial Orders and Chains

Note that maximal elements need not be unique in a partially ordered set, since not all elements need be comparable.

The Axiom of Choice and Its Equivalents

In other words, one can simultaneously choose one element from each set in any non-empty collection of non-empty sets. This principle seems intuitively reasonable, but it is independent of the other axioms of set theory (ZF) and has striking consequences that feel paradoxical — including the existence of non-measurable sets.

These three statements — the Axiom of Choice, Zorn’s Lemma, and the Well-Ordering Principle — are all equivalent, and each implies the others within ZF set theory. The equivalences are proved using transfinite induction. In practice, Zorn’s Lemma is often the most convenient form to apply.

Application: Every Vector Space Has a Basis

As a first application of Zorn’s Lemma, we prove that every vector space — even an infinite-dimensional one — admits a (Hamel) basis. This result is surprising because it is not constructive: one cannot in general write down an explicit basis.

Proof. Let \(\mathcal{S} = \{I \subseteq V : I \text{ is linearly independent}\}\), ordered by inclusion. Let \(\mathcal{C} \subseteq \mathcal{S}\) be an arbitrary chain and define \(Y = \bigcup_{W \in \mathcal{C}} W\). Clearly \(W \subseteq Y\) for all \(W \in \mathcal{C}\), so it suffices to show \(Y \in \mathcal{S}\). Let \(x_1, \ldots, x_n \in Y\) and suppose \(\sum_i \alpha_i x_i = 0\). Each \(x_i\) belongs to some \(W_i \in \mathcal{C}\), and since \(\mathcal{C}\) is totally ordered, the finite collection \(\{W_1, \ldots, W_n\}\) has a maximum \(W^*\). Then \(x_i \in W^* \in \mathcal{S}\) for all \(i\), so the linear independence of \(W^*\) forces \(\alpha_i = 0\) for all \(i\). Thus \(Y\) is linearly independent.

By Zorn’s Lemma, \(\mathcal{S}\) has a maximal element \(M\). We claim \(M\) spans \(V\): if not, there is \(x \in V \setminus \operatorname{span} M\), making \(M \cup \{x\}\) a strictly larger linearly independent set, contradicting maximality. \(\square\)

This proof is a template for many applications of Zorn’s Lemma in functional analysis: define a poset of “partial” objects ordered by extension, show every chain has an upper bound (their union), invoke Zorn to get a maximal object, and show maximality forces the desired global property.

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Chapter 1: Lebesgue Measure

The Lebesgue theory begins by assigning a “size” — a measure — to subsets of \(\mathbb{R}\) in a way that extends the familiar notion of length for intervals, is countably additive, and is well-behaved under limits. The classical Jordan content fails for countable sets; the Lebesgue measure fixes this by allowing countably infinite covers.

Jordan Outer Content and Content

Before defining the Lebesgue measure, it is instructive to understand what the classical theory gets right and where it falls short. The Jordan outer content assigns a size to bounded sets by covering them with finitely many intervals, which works well for “nice” sets like intervals and polygons, but breaks down for more complex sets like the rationals.

1.1 Definition. When \(I\) is a bounded interval \((a,b)\), \([a,b)\), \((a,b]\), or \([a,b]\) with \(a \leq b\), we define \(|I| = b - a\). For unbounded intervals we set \(|I| = \infty\).

\[ c^*(A) = \inf\!\left\{ \sum_{k=1}^{n} |I_k| \;\middle|\; n \in \mathbb{Z}^+,\, \text{each } I_k \text{ is a bounded open interval, and } A \subseteq \bigcup_{k=1}^{n} I_k \right\}. \]

This is the infimum of total lengths of finite open covers.

1.3 Theorem (Properties of Jordan Outer Content). Let \(A, B \subseteq \mathbb{R}\) be bounded.

1. (Translation) \(c^*(a+A) = c^*(A)\) for any \(a \in \mathbb{R}\).
2. (Scaling) \(c^*(rA) = |r|\,c^*(A)\) for \(r \neq 0\).
3. (Inclusion) If \(A \subseteq B\) then \(c^*(A) \leq c^*(B)\).
4. If \(A\) is finite then \(c^*(A) = 0\).
5. If \(I\) is a bounded interval then \(c^*(I) = |I|\).
6. (Subadditivity) \(c^*(A \cup B) \leq c^*(A) + c^*(B)\).
7. \(c^*(\bar{A}) = c^*(A)\).

These properties show that Jordan outer content is well-behaved in many respects. The failure comes when we try to define an interior notion of size: for sets like \(\mathbb{Q} \cap [0,1]\), the outer content is 1 (the rationals are dense), but they should have “content” 0 since they form a negligible set. This motivates the next definition.

1.5 Definition. A bounded set \(A \subseteq \mathbb{R}\) has Jordan content when \(c^*(A) = |I| - c^*(I \setminus A)\) for any interval \(I \supseteq A\), and in that case we write \(c(A) = c^*(A)\). Geometrically, this says the outer content from outside equals the “inner content” from inside.

1.6 Exercise. Show that \(\mathbb{Q} \cap [0,1]\) does not have a well-defined Jordan content (outer content is 1, but any finite cover of the rationals leaves gaps).

1.7 Theorem (Properties of Content). A bounded set \(A\) has Jordan content if and only if \(c^*(A^0) = 0\), where \(A^0\) denotes the boundary of \(A\). Unions, intersections, and differences of sets with content again have content. Every bounded interval has content.

The boundary condition makes intuitive sense: a set has Jordan content precisely when its boundary is “thin” (has zero outer content). The failure of \(\mathbb{Q} \cap [0,1]\) is explained by the fact that its boundary is all of \([0,1]\).

Lebesgue Outer Measure

The key insight of Lebesgue is to replace finite covers by countably infinite covers. This single change allows us to handle countable sets correctly and produces a far richer theory. Every countable set — including \(\mathbb{Q}\) — will now have measure zero, since we can cover the \(n\)-th element with an interval of length \(\varepsilon/2^n\) and make the total as small as we like.

\[ \lambda^*(A) = \inf\!\left\{ \sum_{n=1}^{\infty} |I_n| \;\middle|\; \text{each } I_n \text{ is a bounded open interval and } A \subseteq \bigcup_{n=1}^{\infty} I_n \right\}. \]

The key difference from Jordan content: we allow countably infinite covers.

1.9 Theorem (Properties of Outer Measure). Let \(A, B \subseteq \mathbb{R}\).

1. (Translation) \(\lambda^*(a+A) = \lambda^*(A)\).
2. (Scaling) \(\lambda^*(rA) = |r|\,\lambda^*(A)\) for \(r \neq 0\).
3. (Inclusion) \(A \subseteq B \Rightarrow \lambda^*(A) \leq \lambda^*(B)\).
4. If \(A\) is finite or countable then \(\lambda^*(A) = 0\).
5. If \(I\) is an interval then \(\lambda^*(I) = |I|\).
6. (Countable subadditivity) \(\lambda^*\!\left(\bigcup_{n=1}^\infty A_n\right) \leq \sum_{n=1}^\infty \lambda^*(A_n)\).

Proof sketch for (4): Given \(A = \{a_1, a_2, \ldots\}\) countable and \(\varepsilon > 0\), cover each \(a_n\) by an interval of length \(\varepsilon/2^n\). Then \(\lambda^*(A) \leq \sum \varepsilon/2^n = \varepsilon\); since \(\varepsilon\) was arbitrary, \(\lambda^*(A) = 0\).

Proof sketch for (5): One shows \(\lambda^*(I) \leq |I|\) by taking \(I_1 = (a-\varepsilon, b+\varepsilon)\). For the lower bound, given any countable open cover of a compact subinterval \(K = [a+\varepsilon, b-\varepsilon] \subset I\), extract a finite subcover and chain the overlapping intervals to show the total length is at least \(b-a-2\varepsilon\).

Property (4) is the hallmark of the Lebesgue theory: every countable set is negligible. Property (6) — countable subadditivity — is the best we can hope for in general; the upgrade to countable additivity requires the notion of measurability, which we introduce next.

Lebesgue Measurability

Outer measure is defined for all subsets, but it is only countably additive on a restricted class of sets. The Carathéodory criterion identifies exactly which sets are “well-behaved” enough to serve as the domain of a genuine measure. The idea is that a set \(A\) is measurable if it cleanly splits every other set \(X\) into two pieces whose outer measures add correctly.

\[ \lambda^*(X) = \lambda^*(X \cap A) + \lambda^*(X \setminus A). \]

This is Carathéodory’s criterion: \(A\) measurably splits every set. We let \(\mathcal{M}\) denote the collection of all measurable subsets of \(\mathbb{R}\), and for measurable \(A\) we write \(\lambda(A) = \lambda^*(A)\).

1.11 Note. Since \(X = (X \cap A) \cup (X \setminus A)\), subadditivity gives \(\lambda^*(X) \leq \lambda^*(X \cap A) + \lambda^*(X \setminus A)\) automatically. So measurability requires only the reverse inequality.

This simplification is extremely useful in practice: to prove a set is measurable, we only need to show that splitting by it cannot increase outer measure.

1.12 Theorem (Properties of Measure). The collection \(\mathcal{M}\) is a \(\sigma\)-algebra and the Lebesgue measure \(\lambda : \mathcal{M} \to [0,\infty]\) satisfies:

1. \(\emptyset\) and \(\mathbb{R}\) are measurable.
2. If \(\lambda^*(A) = 0\) then \(A\) is measurable (null sets are measurable).
3. Complements of measurable sets are measurable.
4. Finite unions, intersections, and differences of measurable sets are measurable.
5. Every interval is measurable.
6. Countable unions and intersections of measurable sets are measurable.
7. (Countable additivity) If \(A_1, A_2, \ldots\) are measurable and disjoint, then \(\lambda\!\left(\bigcup_{k=1}^\infty A_k\right) = \sum_{k=1}^\infty \lambda(A_k)\).

\[ \lambda^*(X) = \lambda^*(X \cap A) + \lambda^*((X \setminus A) \cap B) + \lambda^*(X \setminus (A \cup B)) \geq \lambda^*(X \cap (A \cup B)) + \lambda^*(X \setminus (A \cup B)), \]

using subadditivity for the last step since \((X \cap A) \cup ((X \setminus A) \cap B) = X \cap (A \cup B)\).

Countable additivity — property (7) — is what makes the Lebesgue measure genuinely powerful. It means that when we break a measurable set into countably many disjoint pieces, the whole equals the sum of its parts, with no approximation error. This is the precise statement that Jordan content could not achieve for countable collections.

1.13 Corollary (Continuity of Measure). Let \(A_1, A_2, \ldots\) be measurable.

1. If \(A_1 \subseteq A_2 \subseteq \cdots\) (increasing), then \(\lambda\!\left(\bigcup_{n=1}^\infty A_n\right) = \lim_{n\to\infty} \lambda(A_n)\).
2. If \(A_1 \supseteq A_2 \supseteq \cdots\) (decreasing) and \(\lambda(A_m) < \infty\) for some \(m\), then \(\lambda\!\left(\bigcap_{n=1}^\infty A_n\right) = \lim_{n\to\infty} \lambda(A_n)\).

Continuity of measure tells us that \(\lambda\) behaves well under limiting operations on sets, much as a continuous function behaves well under limits of sequences. The finiteness hypothesis in (2) is genuinely necessary: for \(A_n = [n, \infty)\), each \(A_n\) has infinite measure and \(\bigcap A_n = \emptyset\), so the conclusion would fail without it.

1.14 Theorem. All open sets and all closed sets in \(\mathbb{R}\) are measurable. Moreover, every nonempty open set is a countable disjoint union of open intervals, so \(\lambda(U) = \sum_k |I_k|\) for connected components \(I_k\).

1.15 Corollary. \(\lambda^*(A) = \inf\{\lambda(U) \mid U \supseteq A,\, U \text{ open}\}\).

This corollary provides a useful outer approximation: the outer measure of any set can be computed by taking infima over open supersets. A dual inner approximation by closed sets also holds, as we will see in Theorem 1.27.

Non-measurable Sets and Cantor Sets

One might wonder whether every subset of \(\mathbb{R}\) is measurable — would that not simplify everything? The following results show that the situation is more subtle. The measurable sets form a vast collection (even larger in cardinality than the Borel sets), yet non-measurable sets exist, and their construction requires the Axiom of Choice.

1.17 Theorem. \(|\mathcal{M}| = 2^{2^{\aleph_0}}\) — the set of all measurable sets is strictly larger in cardinality than the set of Borel sets (which has cardinality \(2^{\aleph_0}\).

Proof. The standard Cantor set \(C\) has \(\lambda(C) = 0\), so every subset of \(C\) is measurable. Since \(|C| = 2^{\aleph_0}\), the number of subsets of \(C\) is \(2^{2^{\aleph_0}}\).

1.18 Theorem. There exists a non-measurable subset of \(\mathbb{R}\).

Proof (Vitali). Define an equivalence relation on \([0,1]\) by \(x \sim y \Leftrightarrow y - x \in \mathbb{Q}\). Using the Axiom of Choice, select one element from each equivalence class to form \(A \subseteq [0,1]\). Enumerate \(\mathbb{Q} \cap [0,2] = \{a_1, a_2, \ldots\}\) and let \(A_k = a_k + A\). One checks the \(A_k\) are pairwise disjoint and \([1,2] \subseteq \bigcup_k A_k \subseteq [0,3]\). If \(A\) were measurable, countable additivity would give \(\sum_k \lambda(A_k) = \sum_k \lambda(A)\) equal to either 0 or \(\infty\), contradicting \(1 \leq \lambda\!\left(\bigcup_k A_k\right) \leq 3\).

The Vitali construction is a beautiful argument by contradiction: the assumptions of measurability and countable additivity together force an impossibility. It shows that any theory assigning a well-behaved measure to all subsets of \(\mathbb{R}\) is doomed — we must accept that some sets lie outside the domain of \(\lambda\).

σ-Algebras and Borel Sets

Now that we understand what measurability means, it is worth stepping back to examine the abstract structure. The collection \(\mathcal{M}\) is an example of a \(\sigma\)-algebra. Within \(\mathcal{M}\), the Borel sets form a distinguished sub-\(\sigma\)-algebra generated by the open sets — these are the sets reachable by countable operations starting from open sets.

1.19 Notation. We write \(\mathcal{G}\) for the open sets and \(\mathcal{F}\) for the closed sets. For a collection \(\mathcal{C}\) of subsets, \(\mathcal{C}_\sigma\) denotes countable unions of sets from \(\mathcal{C}\), and \(\mathcal{C}_\delta\) denotes countable intersections.

1.20 Definition. A \(\sigma\)-algebra in \(\mathbb{R}\) is a collection \(\mathcal{C}\) of subsets of \(\mathbb{R}\) such that: (1) \(\emptyset \in \mathcal{C}\); (2) \(A \in \mathcal{C} \Rightarrow A^c \in \mathcal{C}\); (3) countable unions of sets in \(\mathcal{C}\) lie in \(\mathcal{C}\).

1.23 Definition. The Borel \(\sigma\)-algebra \(\mathcal{B}\) is the smallest \(\sigma\)-algebra containing all open sets. Elements of \(\mathcal{B}\) are called Borel sets; they include \(\mathcal{G},\ \mathcal{G}_\delta,\ \mathcal{G}_{\delta\sigma},\ldots\) and \(\mathcal{F},\ \mathcal{F}_\sigma,\ \mathcal{F}_{\sigma\delta},\ldots\).

1.25 Theorem. Every Borel set is Lebesgue measurable (\(\mathcal{B} \subseteq \mathcal{M}\), since \(\mathcal{M}\) is itself a \(\sigma\)-algebra containing all open sets.

The containment \(\mathcal{B} \subsetneq \mathcal{M}\) is strict, as Theorem 1.17 shows: there are far more measurable sets than Borel sets. In practice, however, virtually every set encountered in analysis is Borel. The next theorem gives a useful approximation characterization of measurability in terms of Borel sets.

1.27 Theorem. A set \(A \subseteq \mathbb{R}\) is measurable if and only if any of the following equivalent conditions holds:

• For every \(\varepsilon > 0\) there is an open set \(U \supseteq A\) with \(\lambda(U \setminus A) < \varepsilon\).
• There is a \(G_\delta\) set \(B \supseteq A\) with \(\lambda(B \setminus A) = 0\).
• For every \(\varepsilon > 0\) there is a closed set \(K \subseteq A\) with \(\lambda(A \setminus K) < \varepsilon\).
• There is an \(F_\sigma\) set \(C \subseteq A\) with \(\lambda(A \setminus C) = 0\).

This theorem is often called the “regularity” of Lebesgue measure: every measurable set can be approximated from outside by open sets and from inside by closed sets, with the approximation error made arbitrarily small. This tight relationship between topology and measure is one of the special features of the real line.

Density, Category, and the Baire Category Theorem

The concepts in this section provide a topological counterpart to measure-theoretic notions of “smallness.” A set of measure zero is negligible from the measure-theoretic perspective; the analogous topological notion is that of a first-category (meagre) set. Understanding their relationship — and their differences — gives a more complete picture of the structure of \(\mathbb{R}\).

1.29 Definition. A set \(A \subseteq \mathbb{R}\) is dense when \(\bar{A} = \mathbb{R}\), and nowhere dense when every open interval contains a sub-interval disjoint from \(A\) (equivalently, \(\overline{A}^{\,\circ} = \emptyset\).

1.32 Definition. A set is first category (or meagre) if it is a countable union of nowhere dense sets; second category otherwise; residual if its complement is first category.

1.33 Example. Every countable set is first category (each singleton is nowhere dense), so \(\mathbb{Q}\) is first category and \(\mathbb{Q}^c\) is residual.

1.36 Theorem (Baire Category Theorem). (1) Every first-category set has empty interior. (2) Every residual set is dense. (3) A countable intersection of dense open sets is dense.

Proof sketch: If \(A = \bigcup_{k=1}^\infty C_k\) with each \(C_k\) nowhere dense, suppose \(A\) has nonempty interior and choose a closed interval \(I_0 \subseteq A\). Inductively choose nested closed intervals \(I_0 \supseteq I_1 \supseteq I_2 \supseteq \cdots\) with \(I_k \cap C_k = \emptyset\). By the nested interval theorem, \(\bigcap I_k \neq \emptyset\); but any point in the intersection lies in \(I_0 \subseteq A\) yet in no \(C_k\) — contradiction.

The Baire Category Theorem is a fundamental result about the topology of complete metric spaces. It tells us that \(\mathbb{R}\) cannot be written as a countable union of nowhere dense sets — in other words, \(\mathbb{R}\) is topologically “large.” The theorem has far-reaching consequences throughout analysis, including proofs of the open mapping theorem and the uniform boundedness principle.

1.40 Remark. Three notions of “smallness” — countable, measure zero, first category — share the properties that subsets and countable unions of small sets are small, and small sets have empty interior. Yet no containment holds between them in general.

1.41 Theorem. Every subset of \(\mathbb{R}\) is the disjoint union of a set of measure zero and a set of first category.

Proof. Let \(Q = \{a_1, a_2, \ldots\}\). For \(k \in \mathbb{Z}^+\), let \(U_k = \bigcup_{\ell=1}^\infty \left(a_\ell - \frac{1}{2^{k+\ell}}, a_\ell + \frac{1}{2^{k+\ell}}\right)\), so \(\lambda(U_k) \leq 1/2^{k-1}\). Let \(B = \bigcap_{k=1}^\infty U_k\); then \(B\) is residual (countable intersection of dense open sets) yet \(\lambda(B) = 0\). For any \(A\), write \(A = (A \cap B) \cup (A \cap B^c)\): the first part has measure zero and the second is first category (subset of \(B^c\), which is first category).

This striking theorem shows that the two notions of smallness are genuinely different and in some sense complementary: any set can be decomposed into a measure-zero part and a topologically-small part. In particular, there exist sets of measure zero that are topologically large (residual), and sets of first category that have full measure.

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Chapter 2: Lebesgue Integration

Having defined measurable sets, we now define integration for measurable functions. The strategy is: first integrate non-negative simple functions (finite linear combinations of characteristic functions), then non-negative measurable functions (as a supremum over simple functions), and finally general measurable functions (by splitting into positive and negative parts).

Setup: Riemann Integration and Its Limitations

Before building the Lebesgue integral, it is helpful to recall the Riemann theory and understand precisely where it fails. The Riemann integral partitions the domain into intervals; the Lebesgue integral partitions the range instead, which turns out to be far more flexible. The key deficiency of the Riemann theory is its poor behavior under limits: the pointwise limit of Riemann integrable functions need not be Riemann integrable.

2.2–2.3 Definitions. A step function on \([a,b]\) is \(s = \sum_{k=1}^n c_k \chi_{I_k}\) where the \(I_k\) partition \([a,b]\) into intervals; its integral is \(\int_a^b s = \sum c_k |I_k|\). A bounded function \(f: [a,b] \to \mathbb{R}\) is Riemann integrable when the upper and lower Riemann integrals agree: \(U(f) = L(f)\).

2.6 Theorem (Lebesgue’s criterion for Riemann integrability). A bounded function \(f: [a,b] \to \mathbb{R}\) is Riemann integrable if and only if the set of its discontinuities has measure zero.

This theorem gives a clean measure-theoretic characterization of Riemann integrability: a function is Riemann integrable precisely when it is continuous almost everywhere. This also makes it transparent why functions with “too many” discontinuities fail to be Riemann integrable.

2.8 Example. The Dirichlet function \(f = \chi_{\mathbb{Q} \cap [0,1]}\) is discontinuous everywhere and hence not Riemann integrable.

2.11–2.14 Examples (Cantor function). Given a Cantor set \(C = [0,1] \setminus U\), the corresponding Cantor function \(f: [0,1] \to [0,1]\) is continuous and non-decreasing with \(f' = 0\) a.e. on \(U\). When \(C\) is the standard Cantor set, the homeomorphism \(g(x) = x + f(x)\) sends a set of measure zero (\(C\) to a set of measure one, illustrating that continuous images of measurable sets need not be measurable.

The Cantor function is a pathological but illuminating example: it is continuous and non-decreasing, yet its derivative vanishes almost everywhere. Its integral from 0 to 1 is 1, yet integrating its derivative gives 0. This so-called “devil’s staircase” foreshadows the need to be careful about the relationship between differentiation and integration, a theme that will recur throughout the course.

Riemann Integration for Banach-Valued Functions

The Riemann theory of integration extends naturally to functions taking values in a Banach space \(X\). This generalization is not merely formal: in Chapter 6, Fourier analysis on homogeneous Banach spaces requires integrating \(X\)-valued functions, and the convolution \(K * f\) for a scalar kernel \(K \in L^1(\mathbb{T})\) acting on an element \(f\) of a homogeneous Banach space is defined as a Banach-valued Riemann integral. Crucially, the Banach structure — completeness of \(X\) — is exactly what is needed to guarantee that Riemann sums converge. A merely normed, incomplete space would not suffice.

The use of refinements rather than mesh-to-zero is essential: it requires the Riemann sums to stabilize not just along one sequence of finer and finer partitions, but uniformly over all sufficiently fine partitions regardless of sample point placement.

The proof is an immediate application of completeness: the Cauchy condition ensures the net of Riemann sums is a Cauchy net in \(X\), and completeness guarantees convergence to some limit. This is one of many points in the course where the distinction between a normed space and a Banach space carries concrete analytical weight rather than being a formality.

In Chapter 6, the convolution \(K * f\) for \(K \in L^1(\mathbb{T})\) and \(f\) in a homogeneous Banach space is defined precisely as a Banach-valued Riemann integral of the function \(t \mapsto K(t)\, T_t f\), where \(T_t f\) is the translation. The theorem above guarantees this integral is well-defined whenever \(t \mapsto T_t f\) is continuous — which holds in homogeneous spaces by definition.

Measurable Functions

Just as we identified the “right” class of sets (measurable sets) for the domain of \(\lambda\), we now identify the “right” class of functions. A measurable function is one whose preimages of open sets are measurable. This is exactly the requirement that makes the function compatible with the measure structure.

2.15 Definition. The extended reals are \([-\infty, \infty] = \mathbb{R} \cup \{-\infty, \infty\}\) with the natural topology (open balls around \(\pm\infty\) being half-lines).

Working with extended reals allows us to handle limits of sequences of functions without worrying about whether the limit is finite. For instance, the function \(\sum_n f_n\) might be infinite on a set of measure zero; including \(\pm\infty\) as values lets us define the function everywhere while still integrating it sensibly.

2.16 Definition. A function \(f: A \subseteq \mathbb{R} \to [-\infty, \infty]\) is measurable when \(f^{-1}(U)\) is measurable for every open set \(U \subseteq [-\infty, \infty]\). (In particular, the domain \(A\) must be measurable.)

\[ f \text{ is measurable} \iff f^{-1}(a,\infty] \text{ is measurable for all } a \in \mathbb{R} \iff f^{-1}[a,\infty] \text{ is measurable for all } a \in \mathbb{R}. \]

This equivalence is very practical: to check that a function is measurable, it suffices to check that the super-level sets \(\{f > a\}\) are measurable for each real \(a\). This is usually much easier than verifying the full preimage condition.

2.19 Theorem. Characteristic functions, continuous functions, and monotone functions are measurable.

\[ f^+ = \max(f, 0), \quad f^- = \max(-f, 0). \]

Then \(f = f^+ - f^-\) and \(|f| = f^+ + f^-\).

Splitting a function into its positive and negative parts is a convenient bookkeeping device that allows us to reduce questions about general measurable functions to questions about non-negative ones. We will use this decomposition repeatedly in the definition of the Lebesgue integral.

2.21 Theorem. If \(f, g\) are measurable and \(c \in \mathbb{R}\), then \(cf,\ f+g,\ fg,\ |f|,\ f^+,\ f^-\) are all measurable wherever defined.

2.22 Theorem. If \(f_n\) are measurable, then \(\sup_n f_n\), \(\inf_n f_n\), \(\limsup_{n\to\infty} f_n\), and \(\liminf_{n\to\infty} f_n\) are all measurable.

This theorem is crucial: the class of measurable functions is closed under the standard limit operations. In contrast, the Riemann integrable functions fail this closure property — a pointwise limit of Riemann integrable functions need not be Riemann integrable. This is one of the core advantages of the Lebesgue framework.

2.23 Definition. We say a property holds almost everywhere (a.e.) in \(A\) when it holds outside a set of measure zero.

The phrase “almost everywhere” appears constantly in Lebesgue theory. Since sets of measure zero are invisible to the integral, two functions that agree a.e. behave identically for all integration purposes. This motivates the later identification of functions in \(L^p\) spaces.

The Lebesgue Integral

With measurable functions in hand, we build the Lebesgue integral in three steps. First, we integrate simple functions, which are finite linear combinations of characteristic functions. Then we extend to non-negative measurable functions by taking a supremum over simple functions that lie below the function. Finally, we handle general measurable functions using the positive/negative decomposition.

2.25 Definition. A simple function on measurable \(A\) is \(s = \sum_{k=1}^n c_k \chi_{A_k}\), where the \(A_k\) are disjoint measurable sets partitioning \(A\).

\[ \int_A s \,d\lambda = \sum_{k=1}^n c_k \lambda(A_k). \]

The definition for simple functions is exactly what one would expect: multiply the value on each piece by the measure of that piece. Now that we have the integral for simple functions, we can define it for all non-negative measurable functions by approximation.

\[ \int_A f \,d\lambda = \sup\!\left\{ \int_A s \;\middle|\; s \text{ is a non-negative simple function with } s \leq f \right\}. \]

2.28 Note. Any non-negative measurable \(f\) can be approximated from below by an increasing sequence of simple functions: set \(s_n(x) = (k-1)/2^n\) when \((k-1)/2^n \leq f(x) < k/2^n\) for \(k \leq n \cdot 2^n\), and \(s_n(x) = n\) when \(f(x) \geq n\). Then \(s_n \nearrow f\).

This canonical approximation shows that the supremum in the definition of the integral is actually achieved as a limit of an increasing sequence of simple function integrals. The existence of such approximations is what makes the Monotone Convergence Theorem both natural and provable.

The Fundamental Convergence Theorems

The convergence theorems are the heart of the Lebesgue theory. They answer the question: when can we exchange a limit with an integral? The Riemann theory allows this only under very restrictive conditions (uniform convergence). The Lebesgue theory provides far more general and flexible answers, which is one of the primary reasons it superseded the Riemann theory in modern analysis.

\[ \int_A \liminf_{n\to\infty} f_n \leq \liminf_{n\to\infty} \int_A f_n. \]

Proof. It suffices to show that for any non-negative simple \(s \leq \liminf f_n\), we have \(\int_A s \leq \liminf \int_A f_n\). Write \(s = \sum_k a_k \chi_{A_k}\). For any \(0 \leq r < 1\), the sets \(B_{k,n} = \{x \in A_k : f_\ell(x) \geq r a_k \\ \forall \ell \geq n\}\) increase to \(A_k\). Thus \(\int_A f_n \geq \sum_k r a_k \lambda(B_{k,n}) \to r \int_A s\). Since \(r < 1\) was arbitrary, the result follows.

Fatou’s Lemma is a one-sided inequality: the integral of the liminf is at most the liminf of the integrals. The inequality can be strict — consider functions that concentrate their mass near a single point that shifts to infinity. Fatou is often used as a stepping stone toward the stronger results that follow.

\[ \int_A f = \lim_{n\to\infty} \int_A f_n. \]

The Monotone Convergence Theorem says: if functions increase pointwise to a limit, then their integrals converge to the integral of the limit. There is no uniform convergence required, and the limiting integral is allowed to be infinite. This is already a major improvement over the Riemann theory.

\[ \int_A \sum_{n=1}^\infty f_n = \sum_{n=1}^\infty \int_A f_n. \]

This corollary shows that for non-negative functions, integration and infinite summation can always be interchanged. It plays an important role in the proof of Fubini’s theorem and in many other arguments throughout analysis.

\[ \int_A f = \int_A f^+ - \int_A f^-. \]

Equivalently, \(f\) is integrable if and only if \(|f|\) is integrable.

Now we arrive at the most important convergence theorem, which is the workhorse of modern analysis. The key idea is that if all functions in the sequence are bounded in absolute value by a single integrable function, then pointwise convergence implies convergence of integrals.

\[ \lim_{n\to\infty} \int_A f_n = \int_A f. \]

Proof. Apply Fatou’s Lemma to \(g + f_n \geq 0\) to get \(\liminf \int f_n \geq \int f\), and to \(g - f_n \geq 0\) to get \(\limsup \int f_n \leq \int f\).

The Dominated Convergence Theorem is one of the most frequently applied results in all of analysis. The dominating function \(g\) acts as a “budget” for the sequence — as long as no function exceeds this budget, pointwise convergence is enough to guarantee convergence of integrals. This is exactly the kind of result that makes Fourier analysis and the theory of \(L^p\) spaces tractable.

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Chapter 3: The \(L^p\) Spaces

This chapter develops the normed vector spaces \(L^p(A)\) of equivalence classes of measurable functions whose \(p\)th power is integrable, and proves they are Banach spaces (complete normed spaces). The case \(p = 2\) yields a Hilbert space.

Having developed the Lebesgue integral, we now organize integrable functions into structured spaces. The \(L^p\) spaces are the natural setting for functional analysis and are ubiquitous throughout mathematics and its applications. The parameter \(p\) controls the integrability condition: larger \(p\) imposes stronger conditions, and the resulting spaces have finer structure.

Topological Preliminaries

Before studying \(L^p\) spaces specifically, we recall the hierarchy of metric structures. An inner product space has the richest geometry, a normed space less, and a metric space the least. Establishing these foundations carefully allows us to identify which properties of \(L^p\) follow from general principles and which require the specific structure of integration.

3.1–3.4 Definitions. Recall the hierarchy: an inner product on a real vector space \(V\) is a positive-definite symmetric bilinear form \(\langle x, y \rangle\); it induces a norm \(\|x\| = \sqrt{\langle x,x\rangle}\); a norm induces a metric \(d(x,y) = \|x-y\|\); and a metric induces a topology (open sets as unions of open balls).

3.5 Note (Cauchy–Schwarz). Every inner product satisfies \(\langle x, y \rangle \leq \|x\|\|y\|\) and the polarization identity \(\langle x, y \rangle = \tfrac{1}{4}(\|x+y\|^2 - \|x-y\|^2)\).

3.10 Definition. A metric space is complete if every Cauchy sequence converges. A complete normed space is a Banach space; a complete inner-product space is a Hilbert space.

Completeness is the essential property that distinguishes spaces suitable for analysis from mere algebraic constructions. Without completeness, limits of sequences in the space may escape the space itself, making it impossible to define things like Fourier series as convergent infinite sums. Proving that \(L^p\) is complete — the Riesz–Fischer theorem — is therefore one of the central goals of this chapter.

The \(\ell^p\) and \(L^p\) Spaces

We introduce the discrete and continuous \(L^p\) spaces in parallel. The sequence space \(\ell^p\) is a useful finite-dimensional prototype, while \(L^p(A)\) is the main object of study. The passage from \(\ell^p\) to \(L^p\) replaces sums with integrals and requires all the Lebesgue machinery we have developed.

\[ \|x\|_p = \left(\sum_{k=1}^\infty |x_k|^p\right)^{1/p}, \quad \|x\|_\infty = \sup_k |x_k|, \]

and let \(\ell^p = \{x : \|x\|_p < \infty\}\), \(\ell^\infty = \{x : \|x\|_\infty < \infty\}\).

\[ \|f\|_p = \left(\int_A |f|^p\right)^{1/p}, \quad \|f\|_\infty = \inf\{a \geq 0 : \lambda(\{|f| > a\}) = 0\}, \]

and let \(L^p(A) = \{f \in \mathcal{M}(A) : \|f\|_p < \infty\}/{\sim}\) where \(f \sim g \Leftrightarrow f = g\) a.e. (We identify functions equal almost everywhere so that \(\|f\|_p = 0 \Rightarrow f = 0\) in \(L^p\).)

The identification of functions that agree almost everywhere is essential: without it, \(\|\cdot\|_p\) would not be a genuine norm (it would fail to separate points, since any function modified on a null set has the same \(p\)-norm). With this identification, elements of \(L^p\) are equivalence classes of functions rather than individual functions — a subtlety that is important to keep in mind.

3.15 Lemma. If \(f\) is measurable, then \(\{|f| > \|f\|_\infty\}\) has measure zero (the essential supremum is achieved a.e.).

Hölder’s and Minkowski’s Inequalities

To prove that \(L^p\) is a normed space, we need two fundamental inequalities. Hölder’s inequality generalizes the Cauchy–Schwarz inequality and controls the \(L^1\) norm of a product in terms of the \(L^p\) and \(L^q\) norms of the factors. Minkowski’s inequality is simply the triangle inequality for the \(p\)-norm, and its proof uses Hölder in a clever way.

3.16 Definition. We say \(p, q \in [1,\infty]\) are conjugate when \(1/p + 1/q = 1\) (with convention \(1/\infty = 0\).

\[ ab \leq \frac{a^p}{p} + \frac{b^q}{q}. \]

Proof. The area argument: the area of the rectangle \([0,a] \times [0,b]\) is at most the area under \(y = x^{p-1}\) plus the area to its left, giving \(ab \leq \int_0^a x^{p-1}dx + \int_0^b y^{q-1}dy = a^p/p + b^q/q\).

Young’s inequality is a quantitative form of the AM-GM inequality tailored to conjugate exponents. It is the key pointwise estimate used to prove Hölder’s inequality, which then underpins nearly everything else in the \(L^p\) theory.

3.18 Theorem (Hölder’s Inequality). Let \(1/p + 1/q = 1\).

• If \(x \in \ell^p\) and \(y \in \ell^q\), then \(xy \in \ell^1\) with \(\|xy\|_1 \leq \|x\|_p\|y\|_q\).
• If \(f \in L^p(A)\) and \(g \in L^q(A)\), then \(fg \in L^1(A)\) with \(\|fg\|_1 \leq \|f\|_p\|g\|_q\).

Proof. Apply Young’s inequality termwise/pointwise with \(a = |x_k|/\|x\|_p\), \(b = |y_k|/\|y\|_q\), then sum.

3.19 Theorem (Minkowski’s Inequality). For \(p \in [1,\infty]\):

• If \(x, y \in \ell^p\) then \(\|x+y\|_p \leq \|x\|_p + \|y\|_p\).
• If \(f, g \in L^p(A)\) then \(\|f+g\|_p \leq \|f\|_p + \|g\|_p\).

Proof for \(p \in (1,\infty)\): Write \(\|f+g\|^p = \|f+g\| \cdot \|f+g\|^{p-1} \leq (\|f\| + \|g\|)\|f+g\|^{p-1}\), integrate and apply Hölder to each term, then divide both sides by \(\|f+g\|_p^{p-1}\).

3.21 Theorem. \(\ell^p\) and \(L^p(A)\) are normed linear spaces with their \(p\)-norms.

With Hölder and Minkowski established, we have verified that \(L^p(A)\) is a normed vector space. The next step — and the deep result of this chapter — is proving that these spaces are complete.

Completeness of \(L^p\)

3.22 Theorem (Riesz–Fischer). For \(p \in [1,\infty]\), the spaces \(\ell^p\) and \(L^p(A)\) are complete (Banach spaces).

Proof for \(L^p\), \(p < \infty\): Given a Cauchy sequence \(\{f_n\}\), extract a subsequence \(\{f_{n_k}\}\) with \(\|f_{n_{k+1}} - f_{n_k}\|_p \leq 1/2^k\). Define \(g_\ell = \sum_{k=1}^\ell |f_{n_{k+1}} - f_{n_k}|\). By Minkowski, \(\|g_\ell\|_p \leq 1\), and by Fatou, \(\|g\|_p \leq 1\) where \(g = \lim g_\ell\). So \(g < \infty\) a.e., meaning the telescoping series converges a.e. to some limit \(f\). One then uses Fatou again to show \(\|f - f_m\|_p \to 0\).

For \(L^\infty\): A Cauchy sequence in \(L^\infty\) converges uniformly outside a null set, and one defines the limit pointwise off that null set.

The Riesz–Fischer theorem is the cornerstone of functional analysis on \(L^p\) spaces. Its proof is a beautiful interplay between the Lebesgue convergence theorems — particularly Fatou’s lemma — and the algebraic structure of \(L^p\). The fact that completeness is built directly from Fatou’s lemma illustrates how deeply the convergence theorems are woven into the fabric of the theory.

Containment and Separability

3.23 Theorem. Let \(1 \leq p < q \leq \infty\).

1. \(\ell^p \subseteq \ell^q\).
2. If \(\lambda(A) < \infty\), then \(L^q(A) \subseteq L^p(A)\), with \(\|f\|_p \leq \|f\|_q \lambda(A)^{1/p - 1/q}\).

The containment \(L^q \subseteq L^p\) on finite measure spaces means higher integrability implies lower: an \(L^2\) function is automatically in \(L^1\) on a bounded domain.

3.24 Theorem. For \(1 \leq p < q < r \leq \infty\): \(L^p \cap L^r \subseteq L^q \subseteq L^p + L^r\).

3.25 Theorem. For \(1 \leq p < \infty\), \(\ell^p\) and \(L^p[a,b]\) are separable (have a countable dense subset), while \(\ell^\infty\) and \(L^\infty[a,b]\) are not.

The separability of \(L^p[a,b]\) for \(p < \infty\) follows because: step functions with rational coefficients on rational-endpoint intervals are dense in the step functions, which are dense in \(L^p\).

Separability is essential for Fourier analysis: in a separable Hilbert space, one can find a countable orthonormal basis, which is what makes Fourier series representations possible. The failure of separability for \(L^\infty\) corresponds to the fact that the space is “too large” to admit a countable basis in any meaningful sense.

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Chapter 3.5: Egoroff’s Theorem, Lusin’s Theorem, and Fubini’s Theorem

This interlude collects three fundamental results that round out the theory of measurable functions and integration. Littlewood’s three principles — (1) every measurable set is nearly a finite union of intervals, (2) every measurable function is nearly continuous, and (3) every pointwise convergent sequence of measurable functions is nearly uniformly convergent — motivate all three theorems. These results are indispensable for later chapters: Lusin’s theorem gives the density of continuous functions in \(L^p\), and Fubini’s theorem allows us to compute double integrals by iterated integration.

Egoroff’s Theorem

Pointwise convergence and uniform convergence are very different in general: a sequence of functions can converge pointwise everywhere without converging uniformly. Egoroff’s theorem says that on a set of finite measure, pointwise convergence is “nearly” uniform — we can remove a small exceptional set and get uniform convergence on the rest. This is Littlewood’s third principle made precise.

Egoroff’s theorem is remarkable: pointwise convergence, which a priori gives no uniform control, becomes uniform after discarding a set of arbitrarily small measure. The finite measure hypothesis is essential — the result fails on all of \(\mathbb{R}\) (consider \(f_n = \chi_{[n,n+1]}\)). The proof proceeds by noting that for each \(k\), the sets \(A_{n,k} = \{x \in E : |f_m(x) - f(x)| \geq 1/k \text{ for some } m \geq n\}\) decrease to the empty set as \(n \to \infty\), and continuity of measure allows us to choose \(n_k\) so that \(\lambda(A_{n_k, k}) < \varepsilon/2^k\); the union of these exceptional sets has measure less than \(\varepsilon\).

Lusin’s Theorem

While Egoroff’s theorem addresses convergence, Lusin’s theorem addresses the structure of a single measurable function. A measurable function need not be continuous — think of the Dirichlet function — but Lusin’s theorem says it is continuous on most of its domain.

Proof sketch (five steps). (I) Show there is a large \(N\) with \(\lambda\{|f| \geq N\} < \varepsilon/3\), using downward continuity of measure. (II) Approximate \(f\) (on the set \(\{|f| < N\}\)) by a simple function \(\phi\) with \(|\phi - f| < \varepsilon\) pointwise. (III) Approximate the simple function \(\phi\) by a step function \(g\) that agrees with \(\phi\) except on a set of measure at most \(\varepsilon/3\). (IV) Approximate the step function \(g\) by a continuous function \(h\) that agrees with \(g\) except on a set of measure at most \(\varepsilon/3\), by making \(h\) piecewise linear in small neighborhoods of the jump points. (V) Combine the three exceptional sets (each of measure \(\leq \varepsilon/3\)) to get total measure \(\leq \varepsilon\). \(\square\)

Proof (density for \(p < \infty\)). By Lemma 3.5.2, for any measurable \(f\) and any \(\delta > 0\), we can find a continuous \(h\) with \(|f - h| < \delta\) outside an exceptional set \(A\) of measure \(\delta\). Estimating \(\|f - h\|_p^p\) by splitting over \(A\) and \(A^c\) shows \(\|f - h\|_p \to 0\) as \(\delta \to 0\). For unbounded \(f\), first truncate: \(f_n = f \cdot \chi_{\{|f| \leq n\}}\) is bounded, \(f_n \to f\) a.e., and DCT gives \(\|f_n - f\|_p \to 0\).

Failure for \(p = \infty\). Consider \(f = \chi_{[a,(a+b)/2)} - \chi_{[(a+b)/2,b]}\). Any \(g\) equal to \(f\) a.e. is discontinuous (a sequential-continuity argument shows both limits from the left and right at the midpoint must equal the same value, contradicting the values on both sides). Since \(C[a,b]\) is closed in \(L^\infty\) (uniform limits of continuous functions are continuous), no sequence from \(C[a,b]\) can converge to \(f\) in \(L^\infty\). \(\square\)

Lusin’s theorem has the following important corollaries.

Proof. Approximate \(f \in L^p[0,1]\) by a continuous function \(g\) (Lusin), then apply the Weierstrass approximation theorem to approximate \(g\) uniformly by a polynomial. Since \(\lambda[0,1] = 1\), the \(L^p\) norm is bounded by the sup norm. \(\square\)

Fubini’s Theorem

Fubini’s theorem answers the fundamental question: when can a double integral be computed as an iterated integral? It is one of the most used results in all of analysis, appearing throughout the rest of the course in proofs about convolution, the Hilbert transform, and \(L^p\) norms.

The absolute integrability hypothesis \(\iint |f| < \infty\) is crucial: without it, the iterated integrals may exist but disagree (consider \(f(x,y) = (x^2-y^2)/(x^2+y^2)^2\) on \([0,1]^2\)). The companion theorem, Tonelli’s theorem, removes the integrability hypothesis for non-negative functions.

In practice, one typically applies Tonelli’s theorem to \(|f|\) first to verify the integrability hypothesis, then applies Fubini’s theorem to \(f\) itself to exchange order of integration. Fubini’s theorem is used repeatedly in the Fourier analysis chapters: for instance, in proving that the convolution \(f * g\) is well-defined and bounded, and in the \(L^p\) norm formula used in the Hilbert transform argument.

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Chapter 4: Hilbert Spaces

A Hilbert space is a complete inner-product space. The additional structure of an inner product — beyond what a Banach space provides — allows a rich geometry: orthogonal projections, orthonormal bases, and Parseval’s identity. The chief example is \(L^2(A)\).

Among the \(L^p\) spaces, \(L^2\) occupies a special position because \(p = 2\) is the unique value for which the norm arises from an inner product. This inner product structure unlocks an entire geometric theory: we can speak of angles and orthogonality, project onto subspaces, and decompose vectors into orthogonal components. These tools are indispensable for Fourier analysis, where functions are decomposed into orthogonal modes.

Inner Products over \(\mathbb{R}\) and \(\mathbb{C}\)

We now work over both \(\mathbb{R}\) and \(\mathbb{C}\), since complex inner products are essential for Fourier analysis (the complex exponentials \(e^{inx}\) form the natural orthonormal basis). The definition must be modified slightly for \(\mathbb{C}\): linearity in one argument and conjugate-linearity in the other (sesquilinearity) ensures that \(\langle u, u \rangle\) is always real and non-negative.

4.1 Definition. Let \(\mathbb{F} = \mathbb{R}\) or \(\mathbb{C}\). An inner product on a vector space \(W\) over \(\mathbb{F}\) is a map \(\langle \cdot, \cdot \rangle : W \times W \to \mathbb{F}\) satisfying:

1. (Sesquilinearity) \(\langle u+v, w\rangle = \langle u,w\rangle + \langle v,w\rangle\), \(\langle tu, v\rangle = t\langle u,v\rangle\), \(\langle u, tv\rangle = \bar{t}\langle u,v\rangle\).
2. (Conjugate symmetry) \(\langle u,v\rangle = \overline{\langle v,u\rangle}\).
3. (Positive definiteness) \(\langle u,u\rangle \geq 0\) with equality iff \(u = 0\).

4.4 Theorem. Setting \(\|u\| = \sqrt{\langle u,u\rangle}\) gives a norm on \(W\), and the following key identities hold:

• Parallelogram law: \(\|u+v\|^2 + \|u-v\|^2 = 2\|u\|^2 + 2\|v\|^2\).
• Polarization identity (over \(\mathbb{R}\): \(\langle u,v\rangle = \tfrac{1}{4}(\|u+v\|^2 - \|u-v\|^2)\).
• Pythagoras: \(\langle u,v\rangle = 0 \Rightarrow \|u+v\|^2 = \|u\|^2 + \|v\|^2\).
• Cauchy–Schwarz: \(|\langle u,v\rangle| \leq \|u\|\|v\|\), with equality iff \(\{u,v\}\) is linearly dependent.
• Triangle inequality: \(\|u+v\| \leq \|u\| + \|v\|\).

The parallelogram law characterizes inner product spaces among normed spaces: a norm comes from an inner product if and only if the parallelogram law holds. This provides a clean criterion for identifying which Banach spaces are Hilbert spaces. Among the \(L^p\) spaces, only \(L^2\) satisfies the parallelogram law, confirming that \(L^2\) is the unique Hilbert space in this family.

Orthogonality and Gram–Schmidt

Orthogonality is the central geometric concept in Hilbert space theory. Two vectors are orthogonal if their inner product vanishes, generalizing the familiar notion of perpendicularity from Euclidean geometry. The Gram–Schmidt process systematically produces orthonormal sets from arbitrary linearly independent sets, allowing us to work with “nice” coordinates in any Hilbert space.

4.6 Definition. A subset \(A \subseteq W\) is orthogonal if \(\langle u,v\rangle = 0\) for distinct \(u,v \in A\), and orthonormal if additionally \(\|u\| = 1\) for all \(u \in A\).

\[ x = \sum_{k=1}^n \frac{\langle x, u_k\rangle}{\|u_k\|^2} u_k. \]\[ v_n = u_n - \sum_{k=1}^{n-1} \frac{\langle u_n, v_k\rangle}{\|v_k\|^2} v_k \quad (n \geq 2). \]

Then \(\{v_1, v_2, \ldots\}\) is an orthogonal basis with \(\operatorname{span}\{v_1,\ldots,v_n\} = \operatorname{span}\{u_1,\ldots,u_n\}\) for all \(n\).

Gram–Schmidt works perfectly in finite dimensions, but in an infinite-dimensional setting it only produces a countable orthonormal sequence, not necessarily a Hilbert basis. Extending orthonormal sets to full Hilbert bases requires completeness and, in general, Zorn’s lemma. This is where the next section — the projection theorem and orthogonal decomposition — becomes essential.

The Projection Theorem

A key failure of Gram–Schmidt in infinite dimensions: in a non-complete inner product space, a closed subspace need not have orthogonal complement giving a direct sum decomposition. Completeness (the Hilbert space assumption) is essential.

4.27 Theorem (Closest Point / Projection). Let \(H\) be a Hilbert space and \(S \subseteq H\) be closed and convex. Then for every \(a \in H\) there exists a unique \(b \in S\) minimizing \(\|b - a\|\).

\[ \|x_k - x_\ell\|^2 = 2\|x_k-a\|^2 + 2\|x_\ell - a\|^2 - 4\left\|\frac{x_k+x_\ell}{2} - a\right\|^2 \leq 2\|x_k-a\|^2 + 2\|x_\ell-a\|^2 - 4d^2, \]

which goes to zero, so \(\{x_n\}\) is Cauchy. Completeness gives the limit \(b = \lim x_n \in S\) (closed). Uniqueness follows similarly via the parallelogram law.

The projection theorem is a pure consequence of the geometry of inner product spaces: the parallelogram law forces minimizing sequences to be Cauchy, and completeness ensures they converge. The resulting closest-point projection is a fundamental tool in approximation theory and in the decomposition theorem that follows.

4.28 Corollary (Orthogonal Decomposition). Let \(H\) be a Hilbert space and \(U \subseteq H\) a closed subspace. Then:

1. For every \(w \in H\) there exists a unique nearest point \(u \in U\).
2. \(H = U \oplus U^\perp\): every \(w \in H\) writes uniquely as \(w = u + v\) with \(u \in U\), \(v \in U^\perp = \{v : \langle v, u\rangle = 0 \\ \forall u \in U\}\).

Proof. The unique nearest point \(u\) is found by the projection theorem (U is convex and closed). Setting \(v = w - u\), one shows \(v \in U^\perp\): if not, there exists \(u_0 \in U\) with \(\langle v, u_0\rangle > 0\), and then \(\|w - (u + tu_0)\|^2 = \|v\|^2 - 2t\langle v,u_0\rangle + t^2\|u_0\|^2 < \|v\|^2\) for small \(t > 0\), contradicting minimality.

We write \(u = \operatorname{proj}_U(w)\) and \(v = \operatorname{proj}_{U^\perp}(w)\).

Remark. In a non-complete inner product space (e.g., sequences with finitely many nonzero terms, with the \(\ell^2\) inner product), a closed subspace need not satisfy \(W = U \oplus U^\perp\). Completeness is essential.

The orthogonal decomposition \(H = U \oplus U^\perp\) is the infinite-dimensional analogue of the fact that in \(\mathbb{R}^n\), every vector can be written as the sum of its component along a subspace and its component perpendicular to that subspace. This decomposition is the geometric foundation for Fourier series: we will project \(L^2\) functions onto finite-dimensional subspaces spanned by trigonometric polynomials, and the projection is precisely the partial sum of the Fourier series.

Hilbert Bases and Parseval’s Identity

Definition. A Hilbert basis (or complete orthonormal system) for a separable Hilbert space \(H\) is a maximal orthonormal set \(\mathcal{U} = \{u_1, u_2, \ldots\}\).

By Zorn’s lemma, every inner product space has a maximal orthonormal set. In a Hilbert space, maximality is equivalent to density of the span.

4.29 Theorem. Let \(H\) be a separable Hilbert space with countable orthonormal set \(\mathcal{U} = \{u_1, u_2, \ldots\}\). The following are equivalent:

1. \(\mathcal{U}\) is maximal (Hilbert basis).
2. \(\operatorname{span}\mathcal{U}\) is dense in \(H\).
3. For all \(x \in H\): \(x = \sum_{k=1}^\infty \langle x, u_k\rangle u_k\) (convergence in \(H\).
4. Parseval’s identity: For all \(x \in H\): \(\|x\|^2 = \sum_{k=1}^\infty |\langle x, u_k\rangle|^2\).
5. For all \(x, y \in H\): \(\langle x, y\rangle = \sum_{k=1}^\infty \langle x, u_k\rangle \overline{\langle y, u_k\rangle}\).

Note: (4) implies Bessel’s inequality: \(\|x\|^2 \geq \sum_{k=1}^n |\langle x,u_k\rangle|^2\) for any finite orthonormal set.

Parseval’s identity is the infinite-dimensional Pythagorean theorem: the squared norm of a vector equals the sum of the squares of its “Fourier coefficients” \(\langle x, u_k \rangle\). In the context of \(L^2[-\pi,\pi]\) with the trigonometric basis, Parseval’s identity says that the sum of squares of Fourier coefficients equals the integral of the square of the function — a beautiful and useful identity that we will exploit in Chapter 5.

4.30 Theorem. Every separable Hilbert space over \(\mathbb{F}\) is isometrically isomorphic to \(\ell^2(\mathbb{F})\) via the map \(x \mapsto (\langle x, u_1\rangle, \langle x, u_2\rangle, \ldots)\).

This universal classification theorem tells us that, up to isometric isomorphism, there is essentially only one separable Hilbert space: \(\ell^2\). All the different-looking Hilbert spaces we encounter — \(L^2[0,1]\), \(L^2(\mathbb{T})\), Hardy spaces — are all secretly the same abstract object. The Fourier transform, in this light, is simply a concrete realization of this isomorphism.

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Chapter 5: Fourier Analysis

These notes draw from Calvin Kent’s student notes (Lectures 31–36), as Prof. New deferred Fourier series to §§15.1–15.5 and 15.11 of the textbook.

The central question of Fourier analysis is: can every \(L^2\) function be expressed as a series of sines and cosines? The answer is yes, and the tools developed here — the Dirichlet kernel, Fejér kernel, and convolution — also address pointwise and uniform convergence.

The Hilbert space theory of Chapter 4 guarantees the \(L^2\) convergence of Fourier series, but many applications require stronger results: pointwise convergence, or even uniform convergence. The present chapter develops the tools to address these questions. The central objects are the kernels — the Dirichlet kernel for partial sums and the Fejér kernel for Cesàro means — whose analytic properties encode the convergence behavior of the series.

Periodic Functions and Trigonometric Polynomials

A \(2\pi\)-periodic function \(f: \mathbb{R} \to \mathbb{C}\) is determined by its values on \([-\pi, \pi)\), or equivalently on the circle \(\mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}\). We write \(L^p(\mathbb{T})\) or \(L^p[-\pi,\pi]\) for the corresponding \(L^p\) space.

\[ f(x) = a_0 + \sum_{n=1}^m a_n \cos nx + \sum_{n=1}^m b_n \sin nx. \]\[ e^{inx} = \cos nx + i\sin nx, \quad \cos nx = \frac{e^{inx}+e^{-inx}}{2}, \quad \sin nx = \frac{e^{inx}-e^{-inx}}{2i}. \]

The complex notation is more compact and algebraically cleaner. The real and complex forms are completely equivalent: any real trigonometric polynomial can be written as a complex one with conjugate-symmetric coefficients (\(c_{-n} = \overline{c_n}\)), and vice versa. In most theoretical arguments, the complex form is preferred.

Remark 31.2. By the Stone–Weierstrass theorem, the trigonometric polynomials are dense in \(C(\mathbb{T})\) (continuous \(2\pi\)-periodic functions) in the sup norm, and hence also in \(L^p(\mathbb{T})\) for all \(1 \leq p < \infty\).

The density of trigonometric polynomials in \(L^p(\mathbb{T})\) is the key fact that underpins all of Fourier analysis. It guarantees that the trigonometric system, once shown to be orthonormal, is actually a complete orthonormal system — a Hilbert basis for \(L^2\). Without density, we could have a large orthonormal set that still misses some functions.

Fourier Series in \(L^2\)

\[ \|1\|_2^2 = 2\pi, \quad \|\cos nx\|_2^2 = \|\sin nx\|_2^2 = \pi. \]

Normalizing, \(\left\{\frac{1}{\sqrt{2\pi}},\ \frac{1}{\sqrt{\pi}}\cos nx,\ \frac{1}{\sqrt{\pi}}\sin nx\right\}\) is an orthonormal set whose span is dense (the trigonometric polynomials are dense), so it is a Hilbert basis for \(L^2[-\pi,\pi]\).

\[ f(x) = a_0 + \sum_{n=1}^\infty (a_n \cos nx + b_n \sin nx) \quad \text{in } L^2, \]\[ a_0 = \frac{1}{2\pi}\int_{-\pi}^\pi f(x)\,dx, \quad a_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x)\cos nx\,dx, \quad b_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x)\sin nx\,dx. \]\[ f(x) = \sum_{n=-\infty}^\infty c_n e^{inx} \quad \text{in } L^2(\mathbb{T}), \quad \text{where} \quad c_n = \hat{f}(n) = \frac{1}{2\pi}\int_{-\pi}^\pi f(t)e^{-int}\,dt. \]

The \(L^2\) convergence of the Fourier series is a direct consequence of Chapter 4: we have identified a Hilbert basis, and the reconstruction formula follows from the abstract Hilbert space theory. However, \(L^2\) convergence does not immediately tell us what happens at individual points. To study pointwise and uniform convergence, we must analyze the partial sums more concretely using the Dirichlet kernel.

Partial Sums and the Dirichlet Kernel

\[ S_\ell(f)(x) = \sum_{n=-\ell}^\ell c_n e^{inx}. \]\[ S_\ell(f)(x) = \frac{1}{\pi}\int_{-\pi}^\pi f(t)\D_\ell(x-t)\,dt, \]\[ D_\ell(u) = \frac{1}{2}\sum_{n=-\ell}^\ell e^{inu} = \begin{cases} \dfrac{\sin((\ell+\frac{1}{2})u)}{2\sin(\frac{u}{2})} & u \neq 0, \\ \ell + \frac{1}{2} & u = 0. \end{cases} \]

Properties: \(D_\ell\) is real-valued, even, \(2\pi\)-periodic, and \(\int_{-\pi}^\pi D_\ell(u)\,du = \pi\).

The Dirichlet kernel is a convolution kernel: the partial sum \(S_\ell(f)\) is the convolution of \(f\) with \(D_\ell\). The problem with using \(D_\ell\) directly is that it oscillates and takes both positive and negative values, making its behavior difficult to control. In particular, the \(L^1\) norm of \(D_\ell\) grows like \(\log \ell\), which is precisely why Fourier series can fail to converge pointwise for some \(L^1\) functions. The Fejér kernel, introduced next, remedies this by averaging.

Cesàro Means and the Fejér Kernel

The partial sums \(S_\ell(f)\) need not converge pointwise for \(f \in L^1\). Cesàro summation is more robust:

\[ \sigma_m(f)(x) = \frac{1}{m+1}\sum_{\ell=0}^m S_\ell(f)(x) = \frac{1}{\pi}\int_{-\pi}^\pi f(t)\K_m(x-t)\,dt, \]\[ K_m(u) = \frac{1}{m+1}\sum_{\ell=0}^m D_\ell(u) = \begin{cases} \dfrac{1}{2(m+1)}\cdot\dfrac{\sin^2(\frac{(m+1)u}{2})}{\sin^2(\frac{u}{2})} & u \neq 0, \\ \dfrac{m+1}{2} & u = 0. \end{cases} \]

Properties of \(K_m\):

• Real-valued, even, \(2\pi\)-periodic, and non-negative (unlike \(D_\ell\).
• \(\int_{-\pi}^\pi K_m(u)\,du = \pi\).
• \(\max K_m = K_m(0) = (m+1)/2\).
• For fixed \(\delta > 0\): \(K_m(u) \leq \frac{\pi^2}{2(m+1)\sin^2(\delta/2)}\) for \(|u| \geq \delta\), so \(K_m \to 0\) uniformly away from 0.

The convolution can also be written as \(\sigma_m(f)(x) = \frac{1}{\pi}\int_{-\pi}^\pi f(x+t)K_m(t)\,dt\).

The non-negativity of the Fejér kernel is the key property that makes Cesàro means behave better than partial sums. Because \(K_m \geq 0\) and concentrates near 0, convolution with \(K_m\) amounts to taking a weighted average of \(f\) in a neighborhood of \(x\) — and this averaging process is well-suited to producing convergent approximations. The quantitative decay of \(K_m\) away from 0 is precisely what drives the proof of Fejér’s theorem.

The Riemann–Lebesgue Lemma

Theorem 35.1 (Riemann–Lebesgue Lemma). If \(f \in L^1(\mathbb{T})\), then \(\lim_{n \to \infty} c_n(f) = 0\).

\[ |c_n(f)| = |c_n(f-g)| \leq \frac{1}{2\pi}\|f-g\|_1 < \varepsilon. \qquad\square \]

The Riemann–Lebesgue Lemma states that the Fourier coefficients of any \(L^1\) function must decay to zero. Intuitively, the rapidly oscillating functions \(e^{inx}\) become increasingly orthogonal to \(f\) as \(n \to \infty\) — there is increasingly fine cancellation in the integral \(\int f \cdot e^{-inx}\). The proof is a beautiful density argument: we approximate \(f\) by a trigonometric polynomial, for which the result is trivially true, and then control the error.

Fejér’s Theorem

\[ \lim_{m \to \infty} \sigma_m(f)(a) = \frac{f(a^+) + f(a^-)}{2}. \]

Moreover, if \(f\) is continuous on a closed interval \(I\), the convergence is uniform on \(I\).

\[ \sigma_m(f)(a) - \frac{f(a^+)+f(a^-)}{2} = \frac{1}{\pi}\int_0^\pi [(f(a+t)+f(a-t)) - (f(a^+)+f(a^-))]K_m(t)\,dt. \]

Split at \(\delta\): the integral over \([0,\delta]\) is small by continuity of the one-sided limits (so the integrand is at most \(\varepsilon\), using \(\int_0^\pi K_m = \pi/2\). The integral over \([\delta,\pi]\) is small because \(K_m(t) \leq \pi^2/(2(m+1)\sin^2(\delta/2))\) decays as \(m \to \infty\).

Fejér’s theorem is the ideal Fourier convergence result for “mild” functions: wherever one-sided limits exist, the Cesàro means converge to the average of those limits. This is a stronger statement than one might expect from \(L^1\) functions. In particular, at every point of continuity, the Cesàro means converge to the function value — even though the partial sums might not.

Corollary 35.4. If \(f \in L^1(\mathbb{T})\) is continuous at \(a\), then \(\sigma_m(f)(a) \to f(a)\). If \(f\) is continuous everywhere, then \(\sigma_m(f) \to f\) uniformly.

The uniform version is particularly important: it shows that every continuous periodic function is the uniform limit of trigonometric polynomials, providing a constructive proof of the trigonometric case of the Weierstrass approximation theorem.

Applications of Fourier Series

The power of Fourier analysis becomes most visible in its applications. The Fourier series transforms differential equations into algebraic equations mode by mode: each frequency evolves independently, and the full solution is assembled by superposition. We illustrate this with two classical examples.

Forced Damped Oscillator

\[ y_n = A_n \cos nt + B_n \sin nt, \quad A_n = \frac{10-n^2}{(10-n^2)^2+4n^2}, \quad B_n = \frac{2n}{(10-n^2)^2+4n^2}. \]\[ y = e^{-t}(A\cos 3t + B\sin 3t) + \sum_{n=1}^\infty \frac{4}{\pi n^2}(A_n \cos nt + B_n \sin nt). \]

Notice how the Fourier method reduces a linear ODE with a complicated periodic forcing term to a family of simple algebraic equations — one for each frequency. The coefficients \(A_n\) and \(B_n\) are the steady-state response amplitudes, and they decay rapidly in \(n\), ensuring the series converges.

Vibrating String

\[ u(x,t) = \sum_{n=1}^\infty \sin(nx)(A_n \cos nct + B_n \sin nct), \]

where \(A_n\) are the Fourier sine coefficients of \(g\) and \(ncB_n\) are those of \(h\).

The vibrating string is one of the historical motivating problems for Fourier analysis. Each term \(\sin(nx)\cos(nct)\) represents a standing wave — a pure vibrational mode — and the general solution is a superposition of all these modes. The Fourier coefficients of the initial displacement and velocity determine how much energy is initially in each mode, and the modes evolve independently in time. This is the essence of why Fourier analysis is so powerful for partial differential equations.

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Chapter 6: Convergence of Fourier Series

The preceding chapters established that every \(f \in L^2(\mathbb{T})\) has a Fourier series converging to \(f\) in the \(L^2\) norm. But what can we say about pointwise convergence? Uniform convergence? Convergence in \(L^p\) for other values of \(p\)? This chapter addresses all three questions, revealing a rich and sometimes surprising landscape.

Convolution and Summability Kernels

Before addressing convergence, we develop the algebraic framework. The convolution of two functions on the circle is defined by averaging translations of one function against the other. This operation is commutative and associative, and its Fourier transform is the pointwise product of the Fourier transforms — making it a fundamental tool.

The partial sum operator can be expressed as convolution: \(S_N(f)(x) = (f * D_N)(x)\), where \(D_N\) is the Dirichlet kernel. Similarly, the Cesàro mean is \(\sigma_N(f)(x) = (f * F_N)(x)\), where \(F_N\) is the Fejér kernel.

The multiplicativity \(\widehat{f*g}(n) = \hat{f}(n)\hat{g}(n)\) is proved using Fubini’s theorem. It implies that convolution makes \(L^1(\mathbb{T})\) a commutative Banach algebra. However, \(L^1(\mathbb{T})\) has no identity for convolution: any candidate identity \(g\) would need \(\hat{g}(n) = 1\) for all \(n\), contradicting the Riemann–Lebesgue lemma. The Fejér kernels serve as approximate identities.

Proof. Using \(\frac{1}{2\pi}\int K_n = 1\), write \(K_n * f(x) - f(x) = \frac{1}{2\pi}\int K_n(t)[f(x-t) - f(x)]\,dt\). Given \(\varepsilon > 0\), use uniform continuity of \(f\) to find \(\delta > 0\) so \(|t| \leq \delta \Rightarrow |f(x-t)-f(x)| \leq \varepsilon/M\). Split the integral into \([-\delta,\delta]\) and \([\delta, 2\pi-\delta]\). The first piece is at most \(\varepsilon\) by condition (2); the second is at most \(2\|f\|_\infty \cdot \frac{1}{2\pi}\int_\delta^{2\pi-\delta}|K_n| \to 0\) by condition (3). \(\square\)

Proof. Approximate \(f\) in \(L^1\) by some \(g \in C(\mathbb{T})\) with \(\|f - g\|_1 < \varepsilon/M\). Then \(\|K_n * f - f\|_1 \leq \|K_n * (f-g)\|_1 + \|K_n * g - g\|_1 + \|g-f\|_1 \leq M\|f-g\|_1 + \|K_n*g-g\|_\infty + \|f-g\|_1 \to 0\). \(\square\)

Proof. Since \(\hat{f}(n) = 0\) for all \(n\), we have \(\sigma_N(f) = \sum_{n=-N}^N (1-|n|/(N+1))\hat{f}(n)e^{inx} = 0\) for all \(N\). But \(\sigma_N(f) \to f\) in \(L^1\), so \(f = 0\) a.e. \(\square\)

The Dirichlet Kernel and Divergence of Fourier Series

\[ D_N(t) = \sum_{n=-N}^N e^{int} = \frac{\sin((N+\tfrac{1}{2})t)}{\sin(t/2)}. \]

The key fact is that \(\|D_N\|_1 \sim C\log N\), which grows without bound. This growth is the source of divergence phenomena.

Proof. Define bounded linear functionals \(T_N : C(\mathbb{T}) \to \mathbb{C}\) by \(T_N(f) = S_N(f)(0) = \frac{1}{2\pi}\int f(t) D_N(t)\,dt\). Then \(\|T_N\|_{\mathrm{op}} \leq \|D_N\|_1\). In fact \(\|T_N\|_{\mathrm{op}} \geq \frac{1}{2}\|D_N\|_1\): one constructs a sequence \(g_n \in C(\mathbb{T})\) with \(\|g_n\|_\infty = 1\) approximating \(\operatorname{sgn}(D_N)\) closely enough that \(|S_N(g_N)(0)| \geq \frac{1}{2}\|D_N\|_1 \geq C\log N \to \infty\). By the Uniform Boundedness Principle, since \(\sup_N \|T_N\|_{\mathrm{op}} = \infty\), there exists \(f \in C(\mathbb{T})\) such that \(\sup_N |S_N(f)(0)| = \infty\) — i.e., the Fourier series diverges at 0. \(\square\)

Explicit Divergent Construction

The functional analysis argument is non-constructive. We now give an explicit continuous function whose Fourier series diverges at 0, following Felix Zhou’s Ch. 12 construction.

\[ f_n := \sigma_{2n^2}(g_n) = F_{2n^2} * g_n, \]\[ |S_n(f_n)(0)| \geq |S_n(g_n)(0)| - 1 \geq C'\log n \]\[ f(t) = \sum_{k=1}^\infty \frac{1}{k^2} f_{n_k}(n_k t). \]

Since \(\|f_{n_k}\|_\infty \leq 1\), the Weierstrass \(M\)-test gives \(f \in C(\mathbb{T})\). Let \(G_k(t) = f_{n_k}(n_k t)\); the Fourier coefficients of \(G_k\) are supported on \(n_k \mathbb{Z}\), so for \(N < n_k\) the partial sum \(S_N(G_k)\) is the constant \(\hat{G}_k(0) = \hat{f}_{n_k}(0)\).

\[ S_{n_M^2}(f)(0) = \sum_{k=1}^{M-1} \frac{G_k(0)}{k^2} + \frac{S_{n_M}(f_{n_M})(0)}{M^2} + \sum_{k=M+1}^\infty \frac{\hat{f}_{n_k}(0)}{k^2}. \]

Since \(|G_k(0)| \leq 1\) and \(|\hat{f}_{n_k}(0)| \leq 1\), the first and third sums are bounded by \(\sum 1/k^2 < \infty\). The middle term satisfies \(|S_{n_M}(f_{n_M})(0)|/M^2 \geq C'\log(n_M)/M^2 = C' \cdot 3^M\log 2 / M^2 \to \infty\). Therefore \(|S_{n_M^2}(f)(0)| \to \infty\). \(\square\)

Pointwise Convergence for Regular Functions

Despite the existence of divergent Fourier series for continuous functions, there are many classes of functions for which pointwise convergence is guaranteed.

Proof. Using \(\frac{1}{2\pi}\int D_n = 1\), write \(S_n(f)(a) - f(a) = \frac{1}{2\pi}\int [f(a-t) - f(a)] D_n(t)\,dt = \frac{1}{2\pi}\int g(t)\sin(n+\tfrac{1}{2})t\,dt\), where \(g(t) = (f(a-t)-f(a))/\sin(t/2)\). Since \(f\) is differentiable at \(a\), \(g(t) \to -2f'(a)\) as \(t \to 0\), and one shows \(g \in L^1(\mathbb{T})\). The Riemann–Lebesgue lemma then gives \(\int g(t)\sin(n+\tfrac{1}{2})t\,dt \to 0\). \(\square\)

Pointwise Convergence for Discontinuous Functions

The limit is the average of the left and right limits — a beautiful geometric fact. At a point of continuity, this gives \(S_n(f)(a) \to f(a)\).

The condition \(|\hat{f}(n)| = O(1/|n|)\) is satisfied for all functions of bounded variation, in particular for all piecewise Lipschitz functions. The Hardy–Tauberian theorem thus gives a powerful tool for upgrading Cesàro convergence (which Fejér’s theorem provides for functions with one-sided limits) to genuine partial-sum convergence.

The Hilbert Transform and \(L^p\) Convergence

The deepest convergence result is that Fourier series converge in \(L^p\) norm for all \(1 < p < \infty\). The key is the Hilbert transform, which encodes the asymmetry between positive and negative frequencies.

The Hilbert transform multiplies positive Fourier modes by \(-i\) and negative Fourier modes by \(i\). Since \(H\) is linear and orthonormal in Fourier space, Parseval’s theorem immediately gives \(\|H(f)\|_2 \leq \|f\|_2\) — so \(H\) extends to a bounded operator \(H : L^2(\mathbb{T}) \to L^2(\mathbb{T})\).

Proof. (\(\Rightarrow\)) If \(S_N(f) \to f\) in \(L^p\), then \(\sup_N \|S_N(f)\|_p < \infty\) for each \(f\), and the Uniform Boundedness Principle gives a uniform constant \(C\). (\(\Leftarrow\)) If the uniform bound holds, then for any \(f \in L^p\) and \(\varepsilon > 0\), choose a trigonometric polynomial \(p\) with \(\|f - p\|_p < \varepsilon/(C+1)\). Then \(\|S_N(f) - f\|_p \leq \|S_N(f-p)\|_p + \|S_N(p) - p\|_p + \|p - f\|_p \leq (C+1)\|f-p\|_p + \|S_N(p) - p\|_p\). For \(N \geq \deg(p)\), the middle term vanishes. \(\square\)

\[ \int_\mathbb{T} |Hf|^{2k} \leq \sum_{m=1}^k \binom{2k}{2m} \int_\mathbb{T} |f|^{2m}|Hf|^{2k-2m}. \]

An application of Hölder’s inequality with exponents \(k/m\) and \(k/(k-m)\), followed by algebraic manipulation, shows that \(R_k = \|Hf\|_{2k}/\|f\|_{2k}\) satisfies \(R_k^2 \leq 2^{2k}\), giving the desired bound \(C_k = 2^k\). \(\square\)

Proof. For \(p = 2k\) this is Theorem 6.19. For \(p > 2\): choose integer \(k\) with \(p < 2k\). Decompose \(f = f_\alpha + f^\alpha\) where \(f_\alpha = f \cdot \chi_{\{|f| < \alpha/2\}}\) and \(f^\alpha = f \cdot \chi_{\{|f| \geq \alpha/2\}}\). Apply the \(L^{2k}\) bound to \(f_\alpha\) and the \(L^2\) bound to \(f^\alpha\) to get weak-type estimates on the level sets \(\{|Hf| > \alpha\}\). Then use the \(L^p\) norm formula \(\|Hf\|_p^p = p\int_0^\infty \alpha^{p-1}m\{|Hf| > \alpha\}\,d\alpha\) (proved via Fubini), and integrate to obtain \(\|Hf\|_p^p \leq C(p)\|f\|_p^p\).

For \(1 < p < 2\): use the adjoint argument. Define \(H^*\) by \(\int H^*(f)\bar{g} = \int f\overline{H(g)}\) for all \(g \in L^q\) (\(q\) the conjugate exponent). One computes \(\widehat{H^*(f)}(n) = i\operatorname{sgn}(n)\hat{f}(n) = -\widehat{H(f)}(n)\), so \(H^* = -H\), and \(\|H^*\|_{\mathrm{op}, L^p \to L^p} = \|H\|_{\mathrm{op}, L^q \to L^q}\), which is finite since \(q > 2\). \(\square\)

Proof. By Proposition 6.17, \(\|S_N(f)\|_p \leq (1 + \|H\|_{\mathrm{op}})\|f\|_p\). Lemma 6.18 then gives the convergence. \(\square\)

This is one of the deepest results in classical Fourier analysis. It fails at the endpoints \(p = 1\) and \(p = \infty\): the divergence construction of Section 6.2 shows that even for continuous functions (a subset of both \(L^\infty\) and \(L^1\)), Fourier partial sums need not converge pointwise. The special structure of \(L^p\) for \(1 < p < \infty\) — specifically, the reflexivity that allows the adjoint argument — is what makes the result possible.

The boundedness of the Hilbert transform on \(L^p\) is a cornerstone of harmonic analysis, with applications far beyond Fourier series: it appears in the theory of singular integral operators, complex analysis (the Hilbert transform is the boundary value of the conjugate harmonic function), and the study of \(H^p\) Hardy spaces.

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Chapter 7: Which Sequences Are Fourier Coefficients?

The Riemann–Lebesgue Lemma (Chapter 5) tells us that the Fourier coefficients of any \(f \in L^1(\mathbb{T})\) form a sequence in \(c_0(\mathbb{Z}, \mathbb{C})\) — that is, they vanish at infinity. This raises a natural converse question: given an arbitrary sequence \((\beta_n)_{n \in \mathbb{Z}} \in c_0(\mathbb{Z}, \mathbb{C})\), is it the sequence of Fourier coefficients of some \(f \in L^1(\mathbb{T})\)?

The \(L^2\) theory answers a related question completely and affirmatively: by the Riesz–Fischer theorem, a sequence \((\gamma_n)_{n \in \mathbb{Z}}\) is the Fourier coefficient sequence of some \(f \in L^2(\mathbb{T})\) if and only if \((\gamma_n) \in \ell^2(\mathbb{Z}, \mathbb{C})\). The map \(f \mapsto (\hat{f}(n))_{n \in \mathbb{Z}}\) is an isometric isomorphism between \(L^2\) and \(\ell^2\).

For \(L^1\), the question is harder. Since \(L^2 \subsetneq L^1\) (on the compact space \(\mathbb{T}\)) and \(\ell^2 \subsetneq c_0\), the \(L^1\) theory might seem tractable: perhaps every \(c_0\) sequence arises from some \(L^1\) function. The answer — perhaps surprisingly — is no, and the proof is a beautiful application of the Open Mapping Theorem.

The Fourier Transform as a Bounded Operator

Let us package the Fourier transform as a linear map between Banach spaces. Define

\[ \Lambda: (L^1(\mathbb{T}, \mathbb{C}), \|\cdot\|_1) \longrightarrow (c_0(\mathbb{Z}, \mathbb{C}), \|\cdot\|_\infty), \quad \Lambda([f]) = (\hat{f}(n))_{n \in \mathbb{Z}}. \]\[ |\hat{f}(n)| = \left|\int_{\mathbb{T}} f(t) e^{-2\pi i n t}\,dt\right| \leq \int_{\mathbb{T}} |f(t)|\,dt = \|f\|_1, \]

so \(\|\Lambda([f])\|_\infty = \sup_{n \in \mathbb{Z}} |\hat{f}(n)| \leq \|f\|_1\). By the uniqueness theorem (Corollary to Fejér’s theorem), \(\Lambda\) is also injective: if \(\hat{f}(n) = 0\) for all \(n\), then \([f] = [0]\) in \(L^1\).

The question of whether every \(c_0\) sequence is a Fourier coefficient sequence is therefore precisely the question of whether \(\Lambda\) is surjective.

To settle this, we need a theorem from functional analysis — the Open Mapping Theorem — which is itself a consequence of the Baire Category Theorem proved in Chapter 1 of PMATH 450 (or Chapter 8 of PMATH 351).

The Open Mapping Theorem

The key lemma shows that if the image of the closed unit ball under a bounded linear map is “large” (i.e., its closure contains a ball), then we can upgrade closure to actual containment, using completeness.

\[ \left\|y - \sum_{j=1}^n Tx_j\right\| < \frac{1}{2^n}. \]

Since \(\sum_{n=1}^\infty \|x_n\| \leq \sum_{n=1}^\infty m/2^{n-1} = 2m\), and \(X\) is complete, the series \(x = \sum_{n=1}^\infty x_n\) converges to some \(x \in X_{2m}\). By continuity of \(T\), \(Tx = \sum_{n=1}^\infty Tx_n = y\). Thus \(y \in T(X_{2m})\). \(\square\)

\[ B^Y(0, \delta) \subseteq -y + \overline{T(X_m)} \subseteq \overline{T(X_m)} + \overline{T(X_m)} \subseteq \overline{T(X_{2m})}, \]\[ T(X_r) \supseteq Y_{r\delta/8m} \quad \text{for all } r > 0. \]\[ T(G) \supseteq y + T\!\left(B^X(0, \varepsilon)\right) \supseteq y + T(X_{\varepsilon/2}) \supseteq y + Y_{\varepsilon\delta/16m} = B^Y\!\left(y,\, \tfrac{\varepsilon\delta}{16m}\right). \]

Thus every \(y \in T(G)\) is an interior point, so \(T(G)\) is open. \(\square\)

Proof. That \(T^{-1}\) is linear is elementary. If \(G \subseteq X\) is open, then \((T^{-1})^{-1}(G) = T(G)\) is open in \(Y\) by the Open Mapping Theorem. Hence \(T^{-1}\) is continuous. \(\square\)

The Inverse Mapping Theorem is remarkable: it says that if a bounded linear bijection between Banach spaces has a (set-theoretic) inverse, that inverse is automatically bounded. This fails in incomplete normed spaces: one can construct a linear bijection from a normed space to itself whose inverse is unbounded.

The Fourier Map Is Not Surjective

We can now answer the question about Fourier coefficients.

Proof. We established that \(\Lambda\) is continuous, linear, and injective. Suppose for contradiction that \(\Lambda\) were surjective. By the Inverse Mapping Theorem, \(\Lambda^{-1}: c_0(\mathbb{Z}, \mathbb{C}) \to L^1(\mathbb{T})\) would be continuous.

\[ d_N := \Lambda([D_N]) = (\ldots, 0, \underbrace{1, 1, \ldots, 1}_{2N+1 \text{ ones}}, 0, \ldots), \]\[ \|\Lambda^{-1}(d_N)\|_1 = \|D_N\|_1 \to \infty \quad \text{as } N \to \infty. \]

A uniformly bounded sequence \((d_N)\) in \(c_0\) is mapped by \(\Lambda^{-1}\) to an unbounded sequence in \(L^1\). This contradicts the assumed continuity (i.e., boundedness) of \(\Lambda^{-1}\). Therefore \(\Lambda\) is not surjective. \(\square\)

This is a genuinely surprising result. The Riemann–Lebesgue Lemma identifies \(c_0\) as a necessary condition on Fourier coefficients of \(L^1\) functions, but Theorem 7.4 shows that it is far from sufficient.

What Is the Range of the Fourier Transform?

Since \(\Lambda\) is injective but not surjective, its range \(\Lambda(L^1) \subsetneq c_0(\mathbb{Z})\) is a proper subspace. One might guess that the range is \(\ell^1(\mathbb{Z})\) — but this is also wrong. The sequence \(\beta_n = 1/n\) for \(n \geq 1\) and \(\beta_n = 0\) for \(n \leq 0\) lies in \(\ell^2\) (hence is the Fourier coefficient sequence of some \(f \in L^2 \subseteq L^1\)), but \((\beta_n) \notin \ell^1\). So the range of \(\Lambda\) contains sequences outside \(\ell^1\), and yet by Theorem 7.4 it does not contain all of \(c_0\). The range is a proper subset of \(c_0\) that properly contains \(\ell^1\), and its exact description is not known in general.

The question of which sequences are Fourier coefficients is, in general, wide open. Katznelson summarizes the situation with characteristic directness:

The only spaces, defined by conditions of size or smoothness of the functions, for which we obtain a complete characterisation — that is, a necessary and sufficient condition expressed in terms of order of magnitude for a sequence \(\{a_n\}\) to be the Fourier coefficients of a function in the space — are \(L^2(\mathbb{T})\) and its “derivatives” (such as the space of absolutely continuous functions with derivatives in \(L^2\)).

The \(L^2\) case is the Riesz–Fischer theorem. For all other natural function spaces on \(\mathbb{T}\), the problem of characterizing Fourier coefficient sequences remains one of the central open problems of classical harmonic analysis.

The interplay here is beautiful: the Baire Category Theorem — a purely topological result about complete metric spaces — gives us an operator-theoretic tool (the Open Mapping Theorem) that resolves an analytic question (the characterization of Fourier coefficients) by showing a certain operator cannot be surjective. This exemplifies the deep unity of analysis: completeness is not just a technical condition, but a structural property with far-reaching consequences.