PMATH 453: Functional Analysis

Functional analysis is the branch of mathematics concerned with infinite-dimensional vector spaces equipped with analytic structure — norms, inner products, and topologies — together with the linear maps between them. This course develops the three pillars of classical functional analysis: Hilbert space theory (orthogonal decomposition, spectral theory), Banach space theory (the Hahn-Banach theorem, open mapping theorem, closed graph theorem), and general topology (weak topologies, compactness, the Banach-Alaoglu theorem). The prerequisite material from real analysis (PMATH 351) and measure theory (PMATH 450) is reviewed in Chapter 0.

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Chapter 0: Prerequisite Review

Real Analysis (PMATH 351)

Cardinality

Definition 0.1 (Injective, Surjective, Bijective). Let \( f : X \to Y \). We say \( f \) is injective (one-to-one) when for all \( x_1, x_2 \in X \), \( f(x_1) = f(x_2) \implies x_1 = x_2 \). We say \( f \) is surjective (onto) when \( \operatorname{Range}(f) = Y \). We say \( f \) is bijective when it is both injective and surjective.

Definition 0.2 (Cardinality). For sets \( A \) and \( B \):

• We write \( |A| = |B| \) when there exists a bijection \( f : A \to B \).
• We write \( |A| \leq |B| \) when there exists an injection \( f : A \to B \).
• We say \( A \) is countable when \( |A| = |\mathbb{N}| \), writing \( |A| = \aleph_0 \). A set is at most countable when it is finite or countable. A set is uncountable when it is neither finite nor countable.

Theorem 0.3. The sets \( \mathbb{N} \times \mathbb{N} \), \( \mathbb{Z} \), and \( \mathbb{Q} \) are all countable. A countable union of countable sets is countable. The set \( \mathbb{R} \) is uncountable, and \( |\mathbb{R}| = 2^{\aleph_0} \).

Theorem 0.4 (Cantor–Schroeder–Bernstein). If \( |A| \leq |B| \) and \( |B| \leq |A| \) then \( |A| = |B| \).

Theorem 0.5 (Cantor). For every set \( A \), \( |A| < |\mathcal{P}(A)| \). In particular, no set is in bijection with its own power set.

Lebesgue Measure

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

When \( A \) is measurable, its Lebesgue measure is \( \lambda(A) = \lambda^*(A) \).

Theorem 0.8 (Properties of Measure). The collection \( \mathcal{M} \) of measurable subsets of \( \mathbb{R} \) forms a \( \sigma \)-algebra containing all open and closed sets. Measure is countably additive: if \( A_1, A_2, \ldots \) are measurable and pairwise disjoint then \( \lambda(\bigcup_{k=1}^\infty A_k) = \sum_{k=1}^\infty \lambda(A_k) \). If \( A_1 \subseteq A_2 \subseteq \cdots \) then \( \lambda(\bigcup A_n) = \lim_{n\to\infty} \lambda(A_n) \). There exist non-measurable sets.

Example 0.9 (Cantor Set). The standard Cantor set \( C \subseteq [0,1] \) is constructed by iteratively removing open middle thirds. It satisfies \( \lambda(C) = 0 \) and \( |C| = 2^{\aleph_0} \). Every subset of \( C \) is measurable (as a null set). More generally, one can construct a Cantor-like set with any prescribed measure \( m \in [0,1) \).

Functional Analysis Prerequisites (PMATH 450)

The \( L^p \) spaces and their properties, including Hölder’s and Minkowski’s inequalities, are developed in PMATH 450 and recalled below in the Examples section of Chapter 1. The key facts are:

• For a measurable set \( A \subseteq \mathbb{R} \) and \( 1 \leq p \leq \infty \), the spaces \( L^p(A) \) are Banach spaces.
• \( L^2(A) \) is a Hilbert space under \( \langle f, g \rangle = \int_A f \bar{g} \).
• For \( a < b \), \( L^p[a,b] \) is separable for \( 1 \leq p < \infty \), and \( L^\infty[a,b] \) is not separable.

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Chapter 1: Preliminaries

Basic Definitions

We begin by establishing the hierarchy of structured spaces that pervades functional analysis: inner product spaces, normed spaces, metric spaces, and topological spaces. Each layer forgets some structure while retaining key analytic properties.

Definition 1.1 (Inner Product Space). Let \( \mathbb{F} = \mathbb{R} \) or \( \mathbb{C} \). Let \( U \) be a vector space over \( \mathbb{F} \). An inner product on \( U \) is a function \( \langle \cdot, \cdot \rangle : U \times U \to \mathbb{F} \) such that for all \( u, v, w \in U \) and all \( t \in \mathbb{F} \):

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

An inner product space over \( \mathbb{F} \) is a vector space equipped with an inner product. A linear map \( L : U \to V \) between inner product spaces preserves inner product when \( \langle L(x), L(y) \rangle = \langle x, y \rangle \) for all \( x, y \in U \).

Definition 1.2 (Normed Linear Space). A norm on a vector space \( U \) over \( \mathbb{F} \) is a map \( \|\cdot\| : U \to \mathbb{R} \) satisfying for all \( u, v \in U \) and \( t \in \mathbb{F} \):

1. (Scaling) \( \|tu\| = |t|\,\|u\| \).
2. (Positive Definiteness) \( \|u\| \geq 0 \) with \( \|u\| = 0 \iff u = 0 \).
3. (Triangle Inequality) \( \|u+v\| \leq \|u\| + \|v\| \).

A normed linear space is a vector space equipped with a norm. A unit vector is any \( u \) with \( \|u\| = 1 \).

Theorem 1.3. Let \( U \) be an inner product space over \( \mathbb{F} \). Define \( \|u\| = \sqrt{\langle u,u \rangle} \). Then:

1. (Scaling) \( \|tu\| = |t|\,\|u\| \).
2. (Positive Definiteness) \( \|u\| \geq 0 \) with \( \|u\| = 0 \iff u = 0 \).
3. \( \|u+v\|^2 = \|u\|^2 + 2\operatorname{Re}\langle u,v \rangle + \|v\|^2 \).
4. (Pythagoras) If \( \langle u,v \rangle = 0 \) then \( \|u+v\|^2 = \|u\|^2 + \|v\|^2 \).
5. (Parallelogram Law) \( \|u+v\|^2 + \|u-v\|^2 = 2\|u\|^2 + 2\|v\|^2 \).
6. (Polarization Identity) If \( \mathbb{F} = \mathbb{R} \): \( \langle u,v \rangle = \tfrac{1}{4}(\|u+v\|^2 - \|u-v\|^2) \). If \( \mathbb{F} = \mathbb{C} \): \( \langle u,v \rangle = \tfrac{1}{4}(\|u+v\|^2 + i\|u+iv\|^2 - \|u-v\|^2 - i\|u-iv\|^2) \).
7. (Cauchy-Schwarz) \( |\langle u,v \rangle| \leq \|u\|\,\|v\| \), with equality iff \( \{u,v\} \) is linearly dependent.
8. (Triangle Inequality) \( \|u+v\| \leq \|u\| + \|v\| \).

In particular, \( \|\cdot\| \) is a norm on \( U \). Proof omitted.

Definition 1.4 (Metric Space). A metric on a nonempty set \( X \) is a function \( d : X \times X \to \mathbb{R} \) satisfying for all \( x, y, z \in X \):

1. (Positive Definiteness) \( d(x,y) \geq 0 \) with \( d(x,y) = 0 \iff x = y \).
2. (Symmetry) \( d(x,y) = d(y,x) \).
3. (Triangle Inequality) \( d(x,z) \leq d(x,y) + d(y,z) \).

Definition 1.5 (Topology). A topology on a set \( X \) is a collection \( \mathcal{T} \) of subsets of \( X \) such that: (1) \( \emptyset, X \in \mathcal{T} \); (2) finite intersections of elements of \( \mathcal{T} \) lie in \( \mathcal{T} \); (3) arbitrary unions of elements of \( \mathcal{T} \) lie in \( \mathcal{T} \). A subset \( A \subseteq X \) is open when \( A \in \mathcal{T} \) and closed when \( X \setminus A \in \mathcal{T} \).

Note 1.6. These structures form a hierarchy: an inner product induces a norm via \( \|x\| = \sqrt{\langle x,x \rangle} \); a norm on \( V \) induces a metric on any \( X \subseteq V \) via \( d(x,y) = \|x-y\| \); a metric induces a topology by declaring \( A \) open when for every \( a \in A \) there exists \( r > 0 \) with \( B(a,r) \subseteq A \).

\[ \forall \varepsilon > 0 \; \exists n_0 \in \mathbb{Z}^+ \; \forall k \geq n_0 : d(x_k, a) < \varepsilon. \]

The sequence is Cauchy when \( \forall \varepsilon > 0 \; \exists n_0 \; \forall k, \ell \geq n_0 : d(x_k, x_\ell) < \varepsilon \). Every convergent sequence is Cauchy.

Definition 1.8 (Complete Spaces). A metric space is complete when every Cauchy sequence converges. A complete normed linear space is called a Banach space. A complete inner product space is called a Hilbert space.

Definition 1.9 (Dense and Separable). A subset \( A \subseteq X \) is dense in \( X \) when \( \overline{A} = X \). A metric space is separable when it contains a countable dense subset.

Examples of Banach and Hilbert Spaces

Example 1.10 (\( \mathbb{F}^n \). The standard inner product on \( \mathbb{F}^n \) is \( \langle x, y \rangle = y^* x = \sum_{k=1}^n x_k \overline{y_k} \). This induces the 2-norm \( \|x\|_2 = (\sum_{k=1}^n |x_k|^2)^{1/2} \). The space \( \mathbb{F}^n \) is a finite-dimensional separable Hilbert space.

Example 1.11 (\( \ell^2 \). Let \( \ell^2 = \ell^2(\mathbb{F}) = \{ x \in \mathbb{F}^\omega \mid \sum_{k=1}^\infty |x_k|^2 < \infty \} \). The standard inner product is \( \langle x, y \rangle = \sum_{k=1}^\infty x_k \overline{y_k} \), inducing the 2-norm \( \|x\|_2 = (\sum_{k=1}^\infty |x_k|^2)^{1/2} \). The space \( \ell^2 \) is an infinite-dimensional separable Hilbert space.

\[ L^2(A) = L^2(A, \mathbb{F}) = \left\{ f \in M(A) \Big| \int_A |f|^2 < \infty \right\} \big/ \sim \]

where \( f \sim g \iff f = g \) a.e. The standard inner product is \( \langle f, g \rangle = \int_A f \bar{g} \). For \( a < b \), \( L^2[a,b] \) is an infinite-dimensional separable Hilbert space.

Example 1.13 (\( p \)-norms on \( \mathbb{F}^n \). For \( x \in \mathbb{F}^n \), define the \( p \)-norm \( \|x\|_p = (\sum_{k=1}^n |x_k|^p)^{1/p} \) for \( 1 \leq p < \infty \), and \( \|x\|_\infty = \max_k |x_k| \) (the supremum norm). Each gives a norm, and \( \mathbb{F}^n \) is a finite-dimensional separable Banach space under each \( p \)-norm.

Example 1.14 (\( \ell^p \) spaces). For \( x \in \mathbb{F}^\omega \), define \( \|x\|_p = (\sum_{k=1}^\infty |x_k|^p)^{1/p} \) for \( 1 \leq p < \infty \) and \( \|x\|_\infty = \sup_k |x_k| \). Let \( \ell^p = \{ x \in \mathbb{F}^\omega \mid \|x\|_p < \infty \} \). Each \( \ell^p \) is a Banach space; \( \ell^p \) is separable for \( 1 \leq p < \infty \) but \( \ell^\infty \) is not separable.

Example 1.15 (\( L^p(A) \) spaces). For measurable \( A \subseteq \mathbb{R} \) and \( 1 \leq p < \infty \), the \( p \)-norm of \( f \in M(A) \) is \( \|f\|_p = (\int_A |f|^p)^{1/p} \), and the essential supremum is \( \|f\|_\infty = \inf\{ m \geq 0 \mid |f(x)| \leq m \text{ a.e.}\} \). Setting \( L^p(A) = \{ f \in M(A) \mid \|f\|_p < \infty \} / \sim \) with \( f \sim g \iff f = g \) a.e., each \( L^p(A) \) is a Banach space. For \( a < b \), \( L^p[a,b] \) is separable for \( 1 \leq p < \infty \) but \( L^\infty[a,b] \) is not.

Remark 1.16. The triangle inequality for the \( p \)-norms is Minkowski’s Inequality, often proved using Hölder’s Inequality.

Theorem 1.17 (Hölder’s Inequality). Let \( p, q \in [1,\infty] \) with \( \frac{1}{p} + \frac{1}{q} = 1 \).

1. For all \( x, y \in \mathbb{F}^n \) or \( x, y \in \mathbb{F}^\omega \): \( \|xy\|_1 \leq \|x\|_p \|y\|_q \).
2. For all \( f, g \in M(A) \): \( \|fg\|_1 \leq \|f\|_p \|g\|_q \).

Proof omitted.

Theorem 1.18 (Minkowski’s Inequality). Let \( p \in [1,\infty] \).

1. For all \( x, y \in \mathbb{F}^n \) or \( x, y \in \mathbb{F}^\omega \): \( \|x+y\|_p \leq \|x\|_p + \|y\|_p \).
2. For all \( f, g \in M(A) \): \( \|f+g\|_p \leq \|f\|_p + \|g\|_p \).

Proof omitted.

Example 1.19 (Spaces of Continuous Functions). Let \( X \) be a metric space. Let \( F_b(X) \) be the space of bounded functions \( f : X \to \mathbb{F} \), and \( C_b(X) \) the space of bounded continuous functions. Both are Banach spaces under the supremum norm \( \|f\|_\infty = \sup\{|f(x)| \mid x \in X\} \). When \( X \) is compact, \( C(X) = C_b(X) \) is a Banach space. For \( a < b \), \( C[a,b] \) is separable by the Weierstrass Polynomial Approximation Theorem.

Bounded Linear Operators

Remark 1.20. When \( U \) and \( V \) are normed linear spaces, a linear map \( F : U \to V \) is also called a linear operator. When \( V = \mathbb{F} \), it is called a linear functional.

\[ \|F\| = \sup\{ \|Fx\| \mid x \in U, \|x\| \leq 1 \}. \]\[ \|F\| = \sup\{ \|Fx\| \mid x \in U, \|x\| = 1\} = \inf\{ m \geq 0 \mid \|Fx\| \leq m\|x\| \; \forall x \in U\}, \]

and \( \|Fx\| \leq \|F\|\,\|x\| \) for all \( x \in U \). The space of bounded linear operators \( F : U \to V \) is denoted \( B(U,V) \).

Example 1.22. When \( U \) and \( V \) are non-trivial finite-dimensional inner product spaces over \( \mathbb{R} \) and \( F : U \to V \) is linear, the maximum of \( \|Fx\| \) over the unit ball is attained and equals \( \sqrt{\lambda} \) where \( \lambda \) is the largest eigenvalue of \( F^* F \).

Theorem 1.23. Let \( U \) and \( V \) be normed linear spaces.

1. \( B(U,V) \) is a normed linear space under the operator norm.
2. If \( V \) is a Banach space then \( B(U,V) \) is a Banach space.

Proof. Part 1 is routine. For Part 2, let \( (F_n) \) be Cauchy in \( B(U,V) \). For each \( x \in U \), since \( \|F_k x - F_\ell x\| \leq \|F_k - F_\ell\|\,\|x\| \), the sequence \( (F_n x) \) is Cauchy in \( V \) and converges; define \( Gx = \lim_{n\to\infty} F_n x \). Then \( G \) is linear. Since \( (F_n) \) is Cauchy hence bounded, say \( \|F_n\| \leq M \), we get \( \|Gx\| = \lim \|F_n x\| \leq M\|x\| \), so \( G \in B(U,V) \). For \( \varepsilon > 0 \) choose \( m \) so that \( k, n \geq m \implies \|F_n - F_k\| < \varepsilon \); then for \( n \geq m \) and all \( x \), \( \|(F_n - G)x\| = \lim_k \|F_n x - F_k x\| \leq \varepsilon \|x\| \), so \( F_n \to G \) in \( B(U,V) \). \( \square \)

Definition 1.24 (Lipschitz Continuity). A map \( f : (X, d_X) \to (Y, d_Y) \) is Lipschitz continuous with constant \( \ell \geq 0 \) when \( d_Y(f(x), f(y)) \leq \ell \cdot d_X(x,y) \) for all \( x, y \in X \).

Note 1.25. Lipschitz continuous maps are uniformly continuous; they send convergent sequences to convergent sequences and Cauchy sequences to Cauchy sequences.

Theorem 1.26. For a linear map \( F : U \to V \) between normed linear spaces, the following are equivalent:

1. \( F \) is Lipschitz continuous.
2. \( F \) is continuous at some point \( a \in U \).
3. \( F \) is continuous at \( 0 \).
4. \( F \) is bounded.

In this case, \( \|F\| \) is a Lipschitz constant for \( F \).

Proof. (1)\( \implies \)(2)\( \implies \)(3) are immediate. For (3)\( \implies \)(4): if \( F \) is continuous at \( 0 \), choose \( \delta > 0 \) so that \( \|x\| \leq \delta \implies \|Fx\| \leq 1 \); for \( \|x\| = 1 \) we have \( \|F x\| = \frac{1}{\delta}\|F(\delta x)\| \leq \frac{1}{\delta} \), so \( \|F\| \leq \frac{1}{\delta} \). For (4)\( \implies \)(1): \( \|Fx - Fy\| = \|F(x-y)\| \leq \|F\|\,\|x-y\| \). \( \square \)

Dual Spaces

\[ U^* = B(U, \mathbb{F}) = \{ f : U \to \mathbb{F} \mid f \text{ is linear with } \|f\| < \infty \}. \]

By Theorem 1.23, \( U^* \) is always a Banach space.

Theorem 1.28 (Riesz Representation for \( \ell^p \). Let \( p, q \in [1,\infty] \) with \( \frac{1}{p} + \frac{1}{q} = 1 \).

1. The map \( F : \ell^q \to (\ell^p)^* \) given by \( F(b)(a) = \sum_{k=1}^\infty a_k b_k \) is well-defined, linear, injective, and norm-preserving.
2. When \( p \neq \infty \), \( F \) is also surjective, so \( (\ell^p)^* \cong \ell^q \).

Proof sketch. By Hölder’s Inequality, \( |F(b)(a)| \leq \|a\|_p \|b\|_q \), so \( F(b) \) is bounded with \( \|F(b)\| \leq \|b\|_q \). Norm preservation (equality) is shown by testing on suitable extremal sequences. Surjectivity for \( 1 \leq p < \infty \): given \( f \in (\ell^p)^* \), set \( b_k = f(e_k) \) and verify \( b \in \ell^q \) and \( F(b) = f \) by approximation with finite sums. \( \square \)

Remark 1.29. When \( p = \infty \) and \( q = 1 \), the proof of surjectivity breaks down because finitely-supported truncations do not converge in \( \ell^\infty \). Indeed, \( F : \ell^1 \to (\ell^\infty)^* \) is not surjective (as shown later via the Hahn-Banach Theorem).

Theorem 1.31 (Riesz Representation for \( L^p \). Let \( p, q \in [1,\infty] \) with \( \frac{1}{p} + \frac{1}{q} = 1 \), and let \( A \subseteq \mathbb{R} \) be measurable with \( \lambda(A) > 0 \).

1. The map \( F : L^q(A) \to L^p(A)^* \) given by \( F(g)(f) = \int_A fg \) is well-defined, injective, and norm-preserving.
2. When \( 1 \leq p < \infty \), \( F \) is surjective, so \( L^p(A)^* \cong L^q(A) \).

Uniform Boundedness

Definition 1.32. A subset \( A \) of a metric space \( X \) is nowhere dense when the interior of its closure is empty, i.e., \( \overline{A}^\circ = \emptyset \). Equivalently, every nonempty open ball contains a nonempty open ball disjoint from \( A \).

Definition 1.33. A subset \( A \subseteq X \) is first category (or meagre) when it is a countable union of nowhere dense sets, and second category when it is not first category. A set is residual when its complement is first category. Note: \( \mathbb{Q} \) is first category in \( \mathbb{R} \), and \( \mathbb{R} \setminus \mathbb{Q} \) is residual.

Theorem 1.34 (Baire Category Theorem). Let \( X \) be a complete metric space.

1. Every first category set in \( X \) has empty interior.
2. Every residual set in \( X \) is dense.
3. Every countable union of closed sets with empty interiors has empty interior.
4. Every countable intersection of dense open sets is dense.

Proof sketch. Parts (1) and (2) are equivalent by complementation; (3) and (4) are special cases. For (1): suppose \( A = \bigcup_{n=1}^\infty C_n \) with each \( C_n \) nowhere dense, and assume \( A \) has nonempty interior. Choose an open ball \( B_0 \) inside \( A \), then inductively choose a nested sequence of open balls \( B_n \) with \( \operatorname{diam}(B_n) \to 0 \), \( \overline{B_n} \subseteq B_{n-1} \), and \( \overline{B_n} \cap C_n = \emptyset \). By completeness, the centres of \( B_n \) converge to some \( a \in B_0 \subseteq A \), yet \( a \notin C_n \) for all \( n \), contradicting \( a \in A \). \( \square \)

Definition 1.37 (\( \sigma \)-algebra). A \( \sigma \)-algebra in a set \( X \) is a collection \( \mathcal{C} \) of subsets of \( X \) such that: \( \emptyset \in \mathcal{C} \); if \( A \in \mathcal{C} \) then \( A^c \in \mathcal{C} \); and if \( A_1, A_2, \ldots \in \mathcal{C} \) then \( \bigcup_{k=1}^\infty A_k \in \mathcal{C} \).

Theorem 1.40 (Banach-Steinhaus / Uniform Boundedness Principle). Let \( X \) be a Banach space, \( Y \) a normed linear space, and \( S \) a set of bounded linear maps \( L : X \to Y \). Suppose that for every \( x \in X \) there exists \( m_x \geq 0 \) such that \( \|Lx\| \leq m_x \) for all \( L \in S \). Then there exists \( m \geq 0 \) such that \( \|L\| \leq m \) for all \( L \in S \).

Proof. For each \( n \in \mathbb{Z}^+ \), let \( A_n = \{ x \in X \mid \|Lx\| \leq n \; \forall L \in S\} \). Each \( A_n \) is closed, and by hypothesis \( X = \bigcup_{n=1}^\infty A_n \). By the Baire Category Theorem, some \( A_n \) is not nowhere dense, so contains an open ball \( B(a,r) \). For \( \|x\| < r \), we have \( x + a \in B(a,r) \subseteq A_n \), so \( \|L(x)\| \leq \|L(x+a)\| + \|L(a)\| \leq 2n \). Scaling gives \( \|L\| \leq \frac{2n}{r} \) for all \( L \in S \). \( \square \)

\[ E = \left\{ x \in X \;\Big|\; \limsup_{n\to\infty} \|L_{m,n}(x)\| = \infty \; \forall m \in \mathbb{Z}^+ \right\} \]

is a dense \( G_\delta \) set (hence residual and, by the Baire Category Theorem, dense in \( X \).

Proof sketch. Fix \( m \). The sets \( A_\ell = \{ x \mid \|L_{m,n}(x)\| \leq \ell \; \forall n\} \) are closed. If one were not nowhere dense, the Uniform Boundedness Principle would give a bound on \( \|L_{m,n}\| \), contradicting the hypothesis. So all \( A_\ell \) are nowhere dense, making \( B_m = \bigcup_\ell A_\ell \) first category. Then \( E = X \setminus \bigcup_m B_m \) is a countable intersection of dense open sets, hence dense by Baire. \( \square \)

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Chapter 2: Hilbert Spaces

Review of Inner Product Spaces

Definition 2.1 (Hamel Basis). A (Hamel) basis for a vector space \( V \) over any field \( \mathbb{F} \) is a maximal linearly independent set, or equivalently a linearly independent spanning set. Any two Hamel bases for \( V \) have the same cardinality, which defines the (Hamel) dimension \( \dim(V) \).

Definition 2.2 (Orthogonal and Orthonormal Sets). Let \( V \) be an inner product space. A subset \( B \subseteq V \) is orthogonal when \( \langle u, v \rangle = 0 \) for all distinct \( u, v \in B \), and orthonormal when it is orthogonal and every element has norm 1.

\[ \langle x, u_k \rangle = a_k, \quad \langle x, y \rangle = \sum_{k=1}^n a_k \overline{b_k}, \quad \|x\|^2 = \sum_{k=1}^n |a_k|^2. \]

In particular, \( B \) is linearly independent. Proof omitted.

\[ v_n = u_n - \sum_{k=1}^{n-1} \frac{\langle u_n, v_k \rangle}{\|v_k\|^2} v_k. \]

Then \( B = (v_1, v_2, v_3, \ldots) \) is an orthogonal Hamel basis with \( \operatorname{Span}\{v_1, \ldots, v_n\} = \operatorname{Span}\{u_1, \ldots, u_n\} \) for every \( n \). Proof omitted.

Corollary 2.5. Every inner product space of finite or countable Hamel dimension has an orthonormal Hamel basis.

Corollary 2.6. If \( V \) has finite or countable Hamel dimension and \( U \subseteq V \) is a finite-dimensional subspace, then any orthonormal basis for \( U \) extends to an orthonormal basis for \( V \).

Corollary 2.7. Inner product spaces of finite or countable Hamel dimension are isomorphic (as inner product spaces) iff they have the same Hamel dimension. In particular, \( \dim(U) = n \) implies \( U \cong \mathbb{F}^n \), and \( \dim(U) = \aleph_0 \) implies \( U \cong \mathbb{F}^\infty \).

Corollary 2.8. Every finite-dimensional inner product space is complete. Every inner product space of countable Hamel dimension is not complete.

Definition 2.9 (Direct Sum). For subspaces \( U, V \subseteq W \), write \( W = U \oplus V \) when \( W = U + V \) and \( U \cap V = \{0\} \), so every \( x \in W \) has a unique decomposition \( x = u + v \).

\[ U^\perp = \{ x \in V \mid \langle x, u \rangle = 0 \; \forall u \in U \}. \]

Theorem 2.11. Let \( V \) be an inner product space and \( U \subseteq V \) a subspace. Then:

1. \( U^\perp \) is a subspace.
2. \( U^\perp = \{ x \mid \langle x, u \rangle = 0 \; \forall u \in B\} \) for any basis \( B \) of \( U \).
3. \( U \cap U^\perp = \{0\} \).
4. \( U \subseteq (U^\perp)^\perp \).

When \( U \) is finite-dimensional: additionally \( U \oplus U^\perp = V \) and \( (U^\perp)^\perp = U \). Proof omitted.

Definition 2.12 (Orthogonal Projection). When \( V = U \oplus U^\perp \), the orthogonal projection onto \( U \) is the map \( \operatorname{Proj}_U : V \to U \) sending \( x = u + v \) (with \( u \in U, v \in U^\perp \) to \( \operatorname{Proj}_U(x) = u \).

Theorem 2.13. Under the conditions of Definition 2.12, \( \operatorname{Proj}_U(x) \) is the unique point in \( U \) nearest to \( x \). Proof omitted.

\[ \operatorname{Proj}_U(x) = \sum_{k=1}^n \frac{\langle x, u_k \rangle}{\|u_k\|^2} u_k. \]

Note 2.16. If \( U \subseteq W \) is a subspace, its closure \( \overline{U} \) is also a subspace. Moreover \( \overline{U}^\perp = U^\perp \). This follows because if \( v \in U^\perp \) and \( u \in \overline{U} \) with \( x_n \to u \) in \( U \), then \( \langle v, u \rangle = \lim_n \langle v, x_n \rangle = 0 \).

Closed Subspaces of Hilbert Spaces and Orthogonal Projections

Example 2.17. Infinite-dimensional subspaces can behave differently from finite-dimensional ones. Let \( V = \mathbb{F}^\infty \) (finitely-supported sequences) with the standard inner product, and \( U = \{ a \in \mathbb{F}^\infty \mid \sum_k a_k = 0 \} \). One computes \( U^\perp = \{0\} \), so \( (U^\perp)^\perp = V \neq U \) and \( V \neq U \oplus U^\perp \). The issue is that \( V \) (having countable Hamel dimension) is not complete.

Definition 2.18 (Convex Set). A subset \( S \subseteq V \) is convex when for all \( a, b \in S \) and \( 0 \leq t \leq 1 \), \( a + t(b-a) \in S \).

Theorem 2.19 (Best Approximation in Hilbert Spaces). Let \( H \) be a Hilbert space and \( S \subseteq H \) nonempty, closed, and convex. Then for every \( a \in H \) there exists a unique point \( b \in S \) nearest to \( a \).

\[ \|x_k - x_\ell\|^2 = 2\|x_k-a\|^2 + 2\|x_\ell-a\|^2 - 4\Big\|\tfrac{x_k+x_\ell}{2}-a\Big\|^2 \leq 2\|x_k-a\|^2 + 2\|x_\ell-a\|^2 - 4d^2 \to 0, \]

so \( (x_n) \) is Cauchy, converging to some \( b \in S \) (since \( S \) is closed and \( H \) is complete). Then \( \|b-a\| = \lim \|x_n-a\| = d \). Uniqueness follows similarly from the Parallelogram Law. \( \square \)

Theorem 2.20 (Characterization of Closed Subspaces). Let \( H \) be a Hilbert space and \( U \subseteq H \) a subspace. Then \( U \) is closed if and only if \( H = U \oplus U^\perp \). In this case, \( U^\perp \) is closed, \( (U^\perp)^\perp = U \), and for \( x = u + v \) with \( u \in U, v \in U^\perp \), \( u \) is the unique point in \( U \) nearest \( x \).

Proof sketch. (\( \Leftarrow \) If \( H = U \oplus U^\perp \) and \( x_n \in U \) with \( x_n \to a \), write \( a = u + v \); then \( \|v\|^2 = \langle a, v \rangle = \lim \langle x_n, v \rangle = 0 \), so \( a = u \in U \). (\( \Rightarrow \) If \( U \) is closed, apply Theorem 2.19: for each \( x \in H \), the nearest point \( u \in U \) gives \( v = x - u \in U^\perp \) (verified by a variational argument). Uniqueness follows by the same Parallelogram Law argument. \( \square \)

Definition 2.21. When \( U \) is a closed subspace of a Hilbert space \( H \), the orthogonal projection onto \( U \) is the map \( P : H \to U \) given by \( Px = u \) where \( x = u + v \), \( u \in U \), \( v \in U^\perp \).

Unordered Series

The classical theory of sequences and series extends naturally to uncountable index sets, which is essential for working with general Hilbert bases.

Definition 2.22. A series \( \sum_{k=1}^\infty a_k \) in a normed space \( V \) converges absolutely when \( \sum_{k=1}^\infty \|a_k\| < \infty \), and converges unconditionally when every rearrangement converges.

\[ \forall \varepsilon > 0 \; \exists F \in \operatorname{Fin}(K) \; \forall I \in \operatorname{Fin}(K) \; I \supseteq F \implies \|s_I - s\| < \varepsilon, \]

where \( s_I = \sum_{k \in I} a_k \) for finite \( I \). It converges absolutely when \( \sum_{k \in K} \|a_k\| < \infty \).

Theorem 2.26. If \( (a_k)_{k \in K} \) with each \( a_k \geq 0 \) has \( \sum_{k \in K} a_k < \infty \), then at most countably many \( a_k \) are nonzero.

Proof. For each \( n \), let \( K_n = \{ k \mid a_k \geq \frac{1}{n} \} \). If any \( K_n \) were infinite, the sum would be infinite. Thus every \( K_n \) is finite, and \( \{ k \mid a_k > 0 \} = \bigcup_n K_n \) is at most countable. \( \square \)

Theorem 2.28 (Cauchy Criterion). Let \( (a_k)_{k \in K} \) be an indexed set in a normed space \( X \).

1. If \( \sum_{k \in K} a_k \) converges, it is Cauchy.
2. If \( X \) is complete and the series is Cauchy, it converges.

Formulas Involving Orthonormal Indexed Sets

Definition 2.29. An indexed set \( (u_k)_{k \in K} \) in an inner product space is orthonormal when \( \|u_k\| = 1 \) for all \( k \) and \( \langle u_k, u_\ell \rangle = 0 \) for \( k \neq \ell \).

\[ (1)\; \sum_{k \in K} a_k u_k = x, \qquad (2)\; \sum_{k \in K} |a_k|^2 = \|x\|^2, \qquad (3)\; \sum_{k \in K} a_k \overline{b_k} = \langle x, y \rangle. \]

Theorem 2.31. Let \( (u_k)_{k \in K} \) be orthonormal in a Hilbert space \( H \) and let \( (c_k)_{k \in K} \) be scalars.

1. If \( \sum_{k \in K} c_k u_k \) converges to \( x \), then \( x \in \overline{\operatorname{Span}\, B} \) and \( c_k = \langle x, u_k \rangle \).
2. \( \sum_{k \in K} c_k u_k \) converges in \( H \) if and only if \( \sum_{k \in K} |c_k|^2 < \infty \).

\[ \sum_{k \in K} |\langle x, u_k \rangle|^2 \leq \|x\|^2. \]\[ 0 \leq \|x - w_F\|^2 = \|x\|^2 - \sum_{k \in F} |\langle x, u_k \rangle|^2. \]

Taking the supremum over all finite \( F \) gives the inequality. \( \square \)

\[ Px = \sum_{k \in K} \langle x, u_k \rangle u_k, \]

and \( \|P\| = 1 \).

Proof. By Bessel’s Inequality, \( \sum_k |\langle x, u_k \rangle|^2 \leq \|x\|^2 < \infty \), so by Theorem 2.31(2) the sum converges, and by Theorem 2.31(1) it lies in \( U \). For any \( u_k \), \( \langle Px - x, u_k \rangle = \langle x, u_k \rangle - \langle x, u_k \rangle = 0 \), so \( x - Px \in U^\perp \). Thus \( P \) is the orthogonal projection. Since \( \|Px\|^2 = \sum_k |\langle x, u_k \rangle|^2 \leq \|x\|^2 \), we have \( \|P\| \leq 1 \); and \( P(u_k) = u_k \) gives \( \|P\| \geq 1 \). \( \square \)

Hilbert Bases

Theorem 2.34. Let \( H \) be a Hilbert space and \( B \) an orthonormal set. Then \( B \) is a maximal orthonormal set if and only if \( \overline{\operatorname{Span}\, B} = H \).

Proof. If \( B \) is not maximal, we can add a unit vector \( v \) orthogonal to all of \( B \), and then \( v \notin \overline{\operatorname{Span}\, B} \). Conversely, if \( \overline{\operatorname{Span}\, B} \neq H \), then \( (\overline{\operatorname{Span}\, B})^\perp \neq \{0\} \) and we can add a unit vector, so \( B \) is not maximal. \( \square \)

Theorem 2.35.

1. Every inner product space contains a maximal orthonormal set (by Zorn’s Lemma).
2. In a Hilbert space, any two maximal orthonormal sets have the same cardinality.

Proof of (2) sketch. Let \( B = \{u_k\}_{k \in K} \) and \( C = \{v_\ell\}_{\ell \in L} \) be maximal. For each \( k \in K \), let \( L_k = \{\ell \in L \mid \langle u_k, v_\ell \rangle \neq 0\} \); Bessel’s Inequality gives \( |L_k| \leq \aleph_0 \). Since \( C \) is maximal (hence \( \overline{\operatorname{Span}\, C} = H \), for each \( \ell \in L \) there exists \( k \) with \( \langle u_k, v_\ell \rangle \neq 0 \), so \( L = \bigcup_{k \in K} L_k \). Cardinal arithmetic gives \( |L| \leq |K| \cdot \aleph_0 = |K| \). Symmetry gives \( |K| \leq |L| \). \( \square \)

Definition 2.36. A Hilbert basis (or orthonormal basis) for a Hilbert space \( H \) is a maximal orthonormal set. The (Hilbert) dimension \( \dim H \) is the cardinality of any Hilbert basis.

Theorem 2.37. Let \( H \) be a Hilbert space with orthonormal indexed set \( (u_k)_{k \in K} \) and \( B = \{u_k\} \). The following are equivalent:

1. \( B \) is a Hilbert basis.
2. For every \( x \in H \): \( x = \sum_{k \in K} \langle x, u_k \rangle u_k \) (Fourier expansion).
3. For every \( x \in H \): \( \|x\|^2 = \sum_{k \in K} |\langle x, u_k \rangle|^2 \) (Parseval’s identity).
4. For every \( x, y \in H \): \( \langle x, y \rangle = \sum_{k \in K} \langle x, u_k \rangle \overline{\langle y, u_k \rangle} \).

Theorem 2.38. A Hilbert space \( H \) is separable if and only if its Hilbert basis is at most countable.

Example 2.39-2.40 (\( \ell^2(K) \). For any nonempty set \( K \), define \( \ell^2(K, \mathbb{F}) = \{ (c_k)_{k \in K} \mid \sum_{k \in K} |c_k|^2 < \infty \} \) with inner product \( \langle a, b \rangle = \sum_{k \in K} a_k \overline{b_k} \). The standard basis vectors \( (e_\ell)_{\ell \in K} \) form a Hilbert basis. Any Hilbert space over \( \mathbb{F} \) with \( \dim H = |K| \) is isomorphic to \( \ell^2(K, \mathbb{F}) \). In particular, every separable Hilbert space is isomorphic to \( \ell^2 \); for example, \( L^2[a,b] \cong \ell^2 \).

The Dual Space and the Adjoint Map

Theorem 2.41 (Riesz Representation Theorem for Hilbert Spaces). Let \( H \) be a Hilbert space over \( \mathbb{F} \). The map \( \varphi : H \to H^* \) given by \( \varphi(u)(x) = \langle x, u \rangle \) is a bijective norm-preserving map that is linear when \( \mathbb{F} = \mathbb{R} \) and conjugate-linear when \( \mathbb{F} = \mathbb{C} \).

Proof. For \( u \in H \), write \( \varphi_u = \varphi(u) \). Then \( \varphi_u(u) = \|u\|^2 \) gives \( \|\varphi_u\| \geq \|u\| \), and Cauchy-Schwarz gives \( |\varphi_u(x)| \leq \|x\|\,\|u\| \) so \( \|\varphi_u\| \leq \|u\| \). Hence \( \varphi \) is norm-preserving (thus injective). For surjectivity: let \( f \in H^* \), \( f \neq 0 \). Then \( U = \ker(f) \) is a closed proper subspace of \( H \), so \( U^\perp \neq \{0\} \). Choose \( v \in U^\perp \) with \( \|v\| = 1 \) and set \( u = \overline{f(v)} v \); then \( \varphi_u = f \). \( \square \)

Definition 2.42. Using the bijection \( \varphi \) of Theorem 2.41, we define an inner product on \( H^* \) by \( \langle f, g \rangle_{H^*} = \langle \varphi^{-1}(g), \varphi^{-1}(f) \rangle_H \) (note the reversal for conjugate-linearity).

\[ \langle Fx, y \rangle = \langle x, F^* y \rangle \quad \forall x \in H, y \in K. \]

We have \( \|F^*\| = \|F\| \).

Weak Convergence

Definition 2.45. Let \( V \) be an inner product space and \( (u_n) \) a sequence in \( V \). We say \( u_n \to w \) weakly when \( \langle u_n, x \rangle \to \langle w, x \rangle \) for all \( x \in V \).

Note 2.46. Strong convergence (\( \|u_n - w\| \to 0 \) implies weak convergence, but not conversely. For example, any orthonormal sequence \( (u_n) \) in a Hilbert space converges weakly to \( 0 \) (by Parseval’s identity) but does not converge strongly.

Theorem 2.47. Every bounded sequence in a Hilbert space has a weakly convergent subsequence.

Proof sketch. When \( H \) is separable with dense set \( \{a_1, a_2, \ldots\} \): given a bounded sequence \( (u_n) \), use a diagonal subsequence argument to extract a subsequence \( (v_k) \) such that \( \langle v_k, a_m \rangle \) converges for every \( m \). The functional \( f(x) = \lim_k \langle v_k, x \rangle \) is bounded; by the Riesz Representation Theorem there exists \( w \in H \) with \( f(x) = \langle x, w \rangle \), and \( v_k \to w \) weakly. For non-separable \( H \), the sequence lies in a separable subspace, and the result reduces to the separable case. \( \square \)

The Spectral Theorem for Compact Self-Adjoint Operators

Definition 2.48. A compact operator on a Hilbert space \( H \) is a linear map \( F : H \to H \) that sends weakly convergent sequences to (strongly) convergent sequences: \( u_n \to w \) weakly \( \implies Fu_n \to Fw \) in norm.

Note 2.49. Every compact operator is continuous (since strong convergence implies weak convergence), but not conversely. The identity on an infinite-dimensional Hilbert space is continuous but not compact, since any orthonormal sequence converges weakly to \( 0 \) but not in norm.

Definition 2.50. A self-adjoint operator on \( H \) is a continuous \( F : H \to H \) with \( F^* = F \), i.e., \( \langle Fx, y \rangle = \langle x, Fy \rangle \) for all \( x, y \in H \).

Theorem 2.51. Let \( F \) be a continuous self-adjoint operator on \( H \). Then:

1. \( \langle Fu, u \rangle \in \mathbb{R} \) for all \( u \). In particular, all eigenvalues of \( F \) are real.
2. \( \|F\| = \sup\{ |\langle Fu, u \rangle| \mid \|u\| = 1 \} \). Every eigenvalue satisfies \( |\lambda| \leq \|F\| \).

\[ \langle Fu, v \rangle = \tfrac{1}{4}(\langle F(u+v), u+v \rangle - \langle F(u-v), u-v \rangle) \leq \tfrac{M}{2}(\|u\|^2 + \|v\|^2), \]

so choosing \( v = Fu/\|Fu\| \) (when \( Fu \neq 0 \) gives \( \|Fu\| \leq M \). \( \square \)

Example 2.52. The multiplication operator \( F : L^2[0,1] \to L^2[0,1] \), \( F(f)(x) = xf(x) \), is self-adjoint and continuous with no eigenvalues.

Theorem 2.53. Let \( F \) be a compact self-adjoint operator on a Hilbert space \( H \) with \( F \neq 0 \). Then \( F \) has an eigenvalue \( \lambda \) with \( |\lambda| = \|F\| \).

\[ \|Fu_n - \lambda u_n\|^2 = \|Fu_n\|^2 - 2\lambda\langle Fu_n, u_n \rangle + \lambda^2 \leq \|F\|^2 - 2\lambda\langle Fu_n, u_n \rangle + \lambda^2 \to 0. \]

Since \( F \) is compact and \( (u_n) \) is bounded, a subsequence \( (v_k) \) gives \( Fv_k \to Fw \) weakly. Then \( \lambda v_k \to Fw \) in norm. Applying \( F \): \( F(Fw) = \lambda Fw \), so \( \lambda \) is an eigenvalue with eigenvector \( Fw \). \( \square \)

Note 2.54 (Useful Properties for the Spectral Theorem).

1. For continuous \( F \), the eigenspace \( E_\lambda = \ker(F - \lambda I) \) is closed.
2. For self-adjoint \( F \), eigenspaces for distinct eigenvalues are orthogonal.
3. If \( U \subseteq H \) is a closed subspace, the orthogonal projection \( P \) onto \( U \) is self-adjoint and (when \( U \) is finite-dimensional) compact.
4. If \( \lambda \) is a nonzero eigenvalue of a self-adjoint \( F \) with projection \( P \) onto \( E_\lambda \), then \( \lambda P = FP = PF \).
5. For compact \( F \), any eigenspace \( E_\lambda \) for \( \lambda \neq 0 \) is finite-dimensional (else choose an orthonormal sequence \( (e_n) \subset E_\lambda \); then \( e_n \to 0 \) weakly but \( Fe_n = \lambda e_n \not\to 0 \), contradicting compactness).

Theorem 2.55 (Spectral Theorem for Compact Self-Adjoint Operators). Let \( H \) be a Hilbert space and \( F : H \to H \) a nonzero compact self-adjoint operator. Then:

• The set of nonzero eigenvalues is at most countable.
• Each nonzero eigenspace \( E_{\lambda_k} \) is finite-dimensional.
• If there are finitely many nonzero eigenvalues \( \lambda_1, \ldots, \lambda_n \): \[ F = \sum_{k=1}^n \lambda_k P_{\lambda_k}, \] where \( P_{\lambda_k} \) is the orthogonal projection onto \( E_{\lambda_k} \).
• If there are countably many eigenvalues, they can be arranged as \( \lambda_1, \lambda_2, \ldots \) in nonincreasing order of absolute value, with \( \lambda_n \to 0 \), and \[ F = \sum_{k=1}^\infty \lambda_k P_{\lambda_k} \] in the operator norm topology.

Proof sketch. Non-compactness of infinite-dimensional eigenspaces follows from Note 2.54(5). Using Theorem 2.53 iteratively, one extracts eigenvalues \( \lambda_1, \lambda_2, \ldots \) with \( |\lambda_k| = \|F_k\| \) where \( F_{k+1} = F_k - \lambda_k P_{\lambda_k} \). The eigenvalues are in nonincreasing order. If the process is infinite: suppose \( |\lambda_n| \to r > 0 \); picking unit eigenvectors \( u_n \in E_{\lambda_n} \) (which are mutually orthogonal) yields \( \|Fu_{n_k} - Fu_{n_\ell}\|^2 = \lambda_{n_k}^2 + \lambda_{n_\ell}^2 \geq 2r^2 \), contradicting compactness of \( F \). So \( \lambda_n \to 0 \). The completeness of the spectral expansion follows by showing \( \|F_{n+1}\| = |\lambda_{n+1}| \to 0 \) in operator norm. \( \square \)

---

Chapter 3: Banach Spaces

Finite-Dimensional Normed Linear Spaces

Example 3.1. For non-trivial finite-dimensional inner product spaces \( U, V \) over \( \mathbb{R} \) and a linear map \( F : U \to V \), the maximum of \( \|Fx\| \) on the closed unit ball is attained, and \( \|F\| = \sqrt{\lambda} \) where \( \lambda \) is the largest eigenvalue of \( F^*F \).

Theorem 3.2. Let \( U \) be an \( n \)-dimensional normed linear space over \( \mathbb{R} \), with basis \( \{u_1, \ldots, u_n\} \) and associated isomorphism \( F : \mathbb{R}^n \to U \), \( F(t) = \sum_k t_k u_k \). Then both \( F \) and \( F^{-1} \) are Lipschitz continuous.

Proof. Setting \( M = (\sum_k \|u_k\|^2)^{1/2} \), Cauchy-Schwarz gives \( \|F(t)\| \leq M\|t\| \), so \( F \) is Lipschitz. The map \( G = \|\cdot\| \circ F : \mathbb{R}^n \to \mathbb{R} \) is continuous, so attains its minimum \( m > 0 \) on the compact unit sphere. For all \( t \in \mathbb{R}^n \), \( \|F(t)\| \geq m\|t\| \), which gives \( \|F^{-1}(x)\| \leq \frac{1}{m}\|x\| \) and Lipschitz continuity of \( F^{-1} \). \( \square \)

Corollary 3.3. When \( U \) and \( V \) are finite-dimensional normed spaces, every linear map \( F : U \to V \) is Lipschitz continuous.

Corollary 3.4. On a finite-dimensional vector space, any two norms induce the same topology, and convergence in one norm is equivalent to convergence in any other.

Definition 3.5. For a metric space \( Y \) and \( \emptyset \neq X \subseteq Y \), the distance from \( y \in Y \) to \( X \) is \( d(y, X) = \inf\{d(y,x) \mid x \in X\} \). When \( X \) is compact, the infimum is attained.

Theorem 3.6. Let \( W \) be a normed space and \( U \subseteq W \) finite-dimensional. Then for every \( w \in W \) there exists \( u \in U \) with \( d(w,u) = d(w,U) \).

Lemma 3.7 (Riesz’s Lemma). Let \( W \) be a normed space and \( U \subsetneq W \) a proper closed subspace. For every \( 0 < r < 1 \) there exists \( w \in W \setminus U \) with \( \|w\| = 1 \) and \( d(w, U) \geq r \).

Proof. Choose \( v \in W \setminus U \). Let \( d = d(v, U) > 0 \). Choose \( u \in U \) with \( \|v-u\| < d/r \). Let \( w = (v-u)/\|v-u\| \). For any \( x \in U \), \( \|x - w\| = \|v - u\|^{-1} \cdot \|v - (u + \|v-u\|x)\| \geq d/(d/r) = r \). \( \square \)

Theorem 3.8 (Riesz’s Theorem). A normed linear space \( U \) is finite-dimensional if and only if its closed unit ball is compact.

Proof. If \( U \) is finite-dimensional: via the isomorphism \( F : \mathbb{R}^n \to U \) of Theorem 3.2, the preimage of \( B(0,1) \) under \( F \) is closed and bounded in \( \mathbb{R}^n \), hence compact; since \( F \) is a homeomorphism, \( B(0,1) \) is compact. If \( U \) is infinite-dimensional: inductively apply Lemma 3.7 to produce a sequence \( (u_n) \) with \( \|u_n\| = 1 \) and \( \|u_n - u_k\| \geq \frac{1}{2} \) for \( k < n \), so \( B(0,1) \) has no convergent subsequence. \( \square \)

The Hahn-Banach Theorem

Definition 3.9. A seminorm on a vector space \( W \) is a subadditive homogeneous map \( p : W \to \mathbb{R} \): \( p(x+y) \leq p(x)+p(y) \) and \( p(tx) = |t|p(x) \).

Theorem 3.10 (Hahn-Banach, Real Version). Let \( W \) be a real vector space, \( U \subseteq W \) a subspace, \( p : W \to \mathbb{R} \) subadditive and positively homogeneous. Every linear \( f : U \to \mathbb{R} \) with \( f(x) \leq p(x) \) for all \( x \in U \) extends to a linear \( g : W \to \mathbb{R} \) with \( g(x) \leq p(x) \) for all \( x \in W \).

Proof sketch. One shows that an extension by one dimension is always possible: for \( w \in W \setminus U \) and \( V = U + \operatorname{Span}\{w\} \), the value \( r = g(w) \) must be chosen to satisfy \( -p(-y-w) - f(y) \leq r \leq p(x+w) - f(x) \) for all \( x, y \in U \); subadditivity ensures this interval is nonempty. An application of Zorn’s Lemma on the poset of dominated extensions then produces a maximal (hence total) extension. \( \square \)

Theorem 3.11 (Hahn-Banach, Complex Version). Let \( W \) be a vector space over \( \mathbb{F} \), \( U \subseteq W \) a subspace, and \( p \) a seminorm on \( W \). Every linear \( f : U \to \mathbb{F} \) with \( |f(x)| \leq p(x) \) for all \( x \in U \) extends to a linear \( g : W \to \mathbb{F} \) with \( |g(x)| \leq p(x) \) for all \( x \in W \).

Proof. The real case is Theorem 3.10. For \( \mathbb{F} = \mathbb{C} \): write \( f = u + iv \) where \( u, v : U \to \mathbb{R} \). Note \( f(x) = u(x) - iu(ix) \). Extend \( u \) to \( w : W \to \mathbb{R} \) by the real theorem, then set \( g(x) = w(x) - iw(ix) \). One verifies \( g \) is \( \mathbb{C} \)-linear and \( |g(x)| = \operatorname{Re}(g(e^{-i\theta}x)) = w(e^{-i\theta}x) \leq p(e^{-i\theta}x) = p(x) \). \( \square \)

Theorem 3.12 (Hahn-Banach for Bounded Functionals). Let \( W \) be a normed space and \( U \subseteq W \) a subspace. Every \( f \in U^* \) extends to \( g \in W^* \) with \( \|g\| = \|f\| \).

Proof. Apply Theorem 3.11 with \( p(x) = \|f\|\,\|x\| \). \( \square \)

Corollary 3.13. For any \( 0 \neq w \in W \), there exists \( g \in W^* \) with \( g(w) = \|w\| \) and \( \|g\| = 1 \).

Proof. Define \( f : \operatorname{Span}\{w\} \to \mathbb{F} \) by \( f(tw) = t\|w\| \), so \( \|f\| = 1 \). Extend by Theorem 3.12. \( \square \)

Corollary 3.14. Let \( U \subsetneq W \) be a proper closed subspace and \( w \in W \setminus U \). There exists \( g \in W^* \) with \( \|g\| = 1 \), \( g(w) = d(w, U) \), and \( g(u) = 0 \) for all \( u \in U \).

Corollary 3.15. If \( W^* \) is separable then \( W \) is separable.

Proof. Let \( (f_n) \) be dense in \( W^* \). For each \( n \), pick \( u_n \in W \) with \( \|u_n\| = 1 \) and \( f_n(u_n) > \frac{1}{2}\|f_n\| \). Claim \( \overline{\operatorname{Span}\{u_n\}} = W \): if not, Corollary 3.14 gives \( g \in W^* \) with \( \|g\|=1 \) vanishing on all \( u_n \). Since \( (f_n) \) is dense, choose \( n \) with \( \|f_n - g\| < \frac{1}{3} \). Then \( \frac{1}{3} < \frac{1}{2}\|f_n\| < f_n(u_n) = (f_n-g)(u_n) \leq \|f_n-g\| < \frac{1}{3} \), a contradiction. \( \square \)

Note 3.16. Since \( \ell^1 \) is separable but \( \ell^\infty \) is not, Corollary 3.15 implies \( F : \ell^1 \to (\ell^\infty)^* \) of Theorem 1.28 is not surjective (for if it were, \( (\ell^\infty)^* \cong \ell^1 \) would be separable, forcing \( \ell^\infty \) to be separable).

The Hahn-Banach Separation Theorem

Definition 3.17. A point \( a \in A \) in a real vector space is an internal point of \( A \) when for every \( u \in U \) there exists \( r > 0 \) with \( a + tu \in A \) for all \( t \in (-r,r) \). The set of internal points is the core of \( A \), denoted \( \operatorname{Core}(A) \). The interior of \( A \) is always contained in its core.

\[ p_A(x) = \inf\left\{ r > 0 \;\Big|\; \tfrac{1}{r}x \in A \right\}. \]

Theorem 3.19. The Minkowski functional of a convex set with \( 0 \) in its core is positively homogeneous and subadditive.

Theorem 3.20 (Hahn-Banach Separation Theorem). Let \( U \) be a real vector space and \( A, B \subseteq U \) disjoint nonempty convex sets with \( \operatorname{Core}(A) \neq \emptyset \). Then there exists a nonzero linear \( f : U \to \mathbb{R} \) with \( f(x) \leq f(y) \) for all \( x \in A \), \( y \in B \).

Proof sketch. Let \( C = A - B - a + b \) (translating so \( 0 \in \operatorname{Core}(C) \) and \( b-a \notin C \). The Minkowski functional \( p \) of \( C \) satisfies \( p(b-a) \geq 1 \). Define \( f(t(b-a)) = t \cdot p(b-a) \); this satisfies \( f \leq p \) on \( \operatorname{Span}\{b-a\} \). Extend by Theorem 3.10. For \( x \in A \), \( y \in B \): \( x - y - a + b \in C \) so \( p(x-y-a+b) \leq 1 \leq p(b-a) \), yielding \( f(x) \leq f(y) \). \( \square \)

The Riesz Representation Theorem for \( C[a,b]^* \)

Definition 3.22 (Bounded Variation). For \( f : [a,b] \to \mathbb{R} \) and a partition \( P = (x_0, \ldots, x_n) \), define \( V(f,P) = \sum_{k=1}^n |f(x_k) - f(x_{k-1})| \). The total variation is \( V(f,[a,b]) = \sup_P V(f,P) \). We say \( f \) is of bounded variation when \( V(f,[a,b]) < \infty \); the space is denoted \( BV[a,b] \).

\[ \int_a^b f\,dg = \lim_{\|P\|\to 0} \sum_{k=1}^n f(t_k)(g(x_k) - g(x_{k-1})). \]

One can show this limit exists and satisfies \( |\int_a^b f\,dg| \leq V(g,[a,b]) \cdot \|f\|_\infty \).

\[ L(f) = \int_a^b f\,dg \quad \forall f \in C[a,b]. \]

Proof sketch. Extend \( L \) to \( M \in B[a,b]^* \) with \( \|M\| = \|L\| \). Define \( g(x) = M(s_x) \) where \( s_x \) is the step function \( s_x(t) = \mathbf{1}_{t \leq x} \). Show \( g \in BV[a,b] \) by estimating variation against \( \|M\| \). For continuous \( f \), approximate by step functions \( f_n \) in the supremum norm; then \( M(f) = \lim M(f_n) = \lim \sum f(x_k)(g(x_k)-g(x_{k-1})) = \int_a^b f\,dg \). \( \square \)

The Open Mapping Theorem and the Closed Graph Theorem

Theorem 3.27 (Open Mapping Theorem). Let \( U \) and \( V \) be Banach spaces and \( F \in B(U,V) \) surjective. Then \( F \) is open: for every open \( A \subseteq U \), \( F(A) \) is open in \( V \).

Proof. Step 1: Show that for all \( R > 0 \) there exists \( r > 0 \) with \( B(0,r) \subseteq \overline{F(B(0,R))} \). Since \( V = \bigcup_n \overline{F(B(0,n))} \) and \( V \) is complete, Baire gives some \( \overline{F(B(0,n))} \) with nonempty interior; by scaling \( \overline{F(B(0,1))} \) has nonempty interior. Find \( c, r \) with \( B(c,2r) \subseteq \overline{F(B(0,1))} \); by symmetry \( B(0,r) \subseteq \overline{F(B(0,2))} \).

Step 2: Lift the closure: show \( B(0,r) \subseteq F(B(0,1)) \) by an iterative approximation. Given \( y \in B(0,r) \), find \( x_1 \in B(0,\frac{1}{2}) \) with \( \|y - Fx_1\| < \frac{r}{2} \), then \( x_2 \in B(0,\frac{1}{4}) \) with \( \|y - F(x_1+x_2)\| < \frac{r}{4} \), etc. The series \( u = \sum x_k \) converges in \( U \) (since \( \sum \|x_k\| < 1 \) and \( Fu = y \).

Step 3: For open \( A \subseteq U \) and \( v = Fu \in F(A) \) with \( B(u,R) \subseteq A \), find \( r \) with \( B(0,r) \subseteq F(B(0,R)) \); then \( B(v,r) \subseteq F(A) \). \( \square \)

Definition 3.28 (Equivalent Norms). Two norms on a vector space \( U \) are equivalent when they induce the same topology, i.e., when there exist \( \ell, m > 0 \) with \( \|x\|_2 \leq \ell\|x\|_1 \) and \( \|x\|_1 \leq m\|x\|_2 \) for all \( x \).

Corollary 3.29. Let \( U \) be complete under norms \( \|\cdot\|_1 \) and \( \|\cdot\|_2 \). If \( \|x\|_2 \leq \ell\|x\|_1 \) for all \( x \), then the two norms are equivalent.

Proof. The identity \( I : (U,\|\cdot\|_1) \to (U,\|\cdot\|_2) \) is continuous (bounded) and surjective, hence open by the Open Mapping Theorem. So its inverse is also continuous. \( \square \)

Definition 3.31 (Closed Graph). A linear map \( F : U \to V \) has a closed graph when for every sequence \( (x_n) \), if \( x_n \to a \) in \( U \) and \( Fx_n \to b \) in \( V \), then \( b = Fa \).

Theorem 3.32 (Closed Graph Theorem). Let \( U \) and \( V \) be Banach spaces and \( F : U \to V \) linear. If \( F \) has a closed graph, then \( F \) is continuous (bounded).

Proof. Define a second norm on \( U \) by \( \|x\|_3 = \|x\|_1 + \|Fx\|_2 \). If \( (x_n) \) is Cauchy in \( \|\cdot\|_3 \), it is Cauchy in both \( \|\cdot\|_1 \) and \( \|\cdot\|_2 \), so \( x_n \to a \) and \( Fx_n \to b \). Since \( F \) has closed graph, \( b = Fa \), and \( x_n \to a \) in \( \|\cdot\|_3 \). So \( (U, \|\cdot\|_3) \) is complete. Since \( \|x\|_1 \leq \|x\|_3 \), Corollary 3.29 gives \( \ell \) with \( \|x\|_3 \leq \ell\|x\|_1 \). Then \( \|Fx\|_2 \leq \|x\|_3 \leq \ell\|x\|_1 \), so \( F \) is bounded. \( \square \)

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Chapter 4: Topology

Topological Spaces and Bases

Definition 4.1 (Topology). A topology on a set \( X \) is a collection \( \mathcal{T} \) of subsets of \( X \) (the open sets) such that: (1) \( \emptyset, X \in \mathcal{T} \); (2) \( \mathcal{T} \) is closed under arbitrary unions; (3) \( \mathcal{T} \) is closed under finite intersections. A subset \( A \subseteq X \) is closed when \( A^c \in \mathcal{T} \). The interior \( A^\circ \) is the largest open set contained in \( A \); the closure \( \overline{A} \) is the smallest closed set containing \( A \).

A topology \( \mathcal{S} \) is coarser than \( \mathcal{T} \) (and \( \mathcal{T} \) is finer) when \( \mathcal{S} \subseteq \mathcal{T} \). Given any collection \( \mathcal{S} \) of subsets of \( X \), there is a unique coarsest topology containing \( \mathcal{S} \) (the topology generated by \( \mathcal{S} \), consisting of arbitrary unions of finite intersections of elements of \( \mathcal{S} \).

A basis for a topology on \( X \) is a collection \( \mathcal{B} \) with: (1) \( X = \bigcup \mathcal{B} \); (2) for all \( U, V \in \mathcal{B} \) and \( a \in U \cap V \), there exists \( W \in \mathcal{B} \) with \( a \in W \subseteq U \cap V \).

Theorem 4.2. Let \( \mathcal{B} \) be a basis generating topology \( \mathcal{T} \). Then \( A \in \mathcal{T} \) iff for every \( a \in A \) there exists \( U \in \mathcal{B} \) with \( a \in U \subseteq A \), equivalently iff \( A \) is a union of elements of \( \mathcal{B} \).

Example 4.3. In a metric space \( X \), the collection of open balls \( \{B(a,r) \mid a \in X, r > 0\} \) is a basis for the metric topology.

Theorem 4.4. Let \( X \) be a topological space with basis \( \mathcal{B} \), and \( A \subseteq X \). Then \( a \in \overline{A} \) if and only if \( A \cap U \neq \emptyset \) for every \( U \in \mathcal{B} \) with \( a \in U \).

Example 4.5 (Subspace Topology). When \( X \subseteq Y \) and \( Y \) has topology \( \mathcal{T} \), the subspace topology on \( X \) is \( \{V \cap X \mid V \in \mathcal{T}\} \).

Example 4.6 (Product Topology). For topological spaces \( X, Y \), the product topology on \( X \times Y \) has basis \( \{U \times V \mid U \subseteq X, V \subseteq Y \text{ open}\} \).

Example 4.7 (Quotient Topology). For an equivalence relation \( \sim \) on a topological space \( X \) with quotient map \( q : X \to X/{\sim} \), the quotient topology is \( \{V \subseteq X/{\sim} \mid q^{-1}(V) \text{ open in } X\} \).

Continuous Functions and Compact Sets

Definition 4.8 (Hausdorff). A topological space \( X \) is Hausdorff when for all distinct \( a, b \in X \) there exist disjoint open sets \( U, V \) with \( a \in U \), \( b \in V \). All metric spaces are Hausdorff.

Definition 4.10 (Continuity). A function \( f : X \to Y \) between topological spaces is continuous when \( f^{-1}(V) \) is open in \( X \) for every open \( V \subseteq Y \).

Definition 4.11 (Compactness). A subset \( A \subseteq X \) is compact when every open cover of \( A \) has a finite subcover.

Theorem 4.12. A subset \( A \subseteq X \subseteq Y \) is compact in \( X \) (with the subspace topology) iff it is compact in \( Y \).

Theorem 4.15. A topological space \( X \) is compact iff it has the finite intersection property on closed sets: every collection of closed sets with the property that every finite subcollection has nonempty intersection has nonempty total intersection.

Theorem 4.16. Every closed subspace of a compact space is compact.

Theorem 4.17. Every compact subspace of a Hausdorff space is closed.

Theorem 4.18. The continuous image of a compact space is compact.

Theorem 4.20 (Extreme Value Theorem). A continuous map \( f : X \to \mathbb{R} \) on a compact space attains its maximum and minimum.

Theorem 4.21. Let \( X \) be compact, \( Y \) Hausdorff, and \( f : X \to Y \) continuous and bijective. Then \( f \) is a homeomorphism.

Urysohn’s Lemma and the Tietze Extension Theorem

Definition 4.23 (Normal Space). A topological space is normal when all one-point sets are closed and for all disjoint closed sets \( A, B \) there exist disjoint open sets \( U, V \) with \( A \subseteq U \), \( B \subseteq V \). All metric spaces are normal.

Theorem 4.25 (Urysohn’s Lemma). Let \( X \) be normal and \( A, B \subseteq X \) disjoint and closed. There exists a continuous \( f : X \to [0,1] \) with \( f|_A = 0 \) and \( f|_B = 1 \).

Proof sketch. Enumerate \( [0,1] \cap \mathbb{Q} = \{a_0, a_1, a_2, \ldots\} \) with \( a_0 = 0, a_1 = 1 \). Inductively construct open sets \( U_r \) for each \( r \in [0,1] \cap \mathbb{Q} \) such that \( r < s \implies \overline{U_r} \subseteq U_s \), with \( A \subseteq U_0 \) and \( B \cap U_1 = \emptyset \). Define \( f(x) = \inf\{r \in \mathbb{Q} \mid x \in U_r\} \) and verify continuity by showing preimages of open intervals are open. \( \square \)

Theorem 4.26 (Tietze Extension Theorem). Let \( X \) be normal, \( A \subseteq X \) closed, and \( a < b \).

1. Every continuous \( f : A \to [a,b] \) extends to continuous \( g : X \to [a,b] \).
2. Every continuous \( f : A \to (a,b) \) extends to continuous \( g : X \to (a,b) \).

Proof sketch of (1). WLOG \( [a,b] = [-1,1] \). Inductively, apply Urysohn’s Lemma to construct a continuous approximation \( g_1 : X \to [-\frac{1}{3}, \frac{1}{3}] \) with \( \|f - g_1|_A\|_\infty \leq \frac{2}{3} \). Repeat on the residual \( f - g_1|_A \) scaled by \( \frac{2}{3} \), obtaining \( g_2 \) with \( \|g_k\|_\infty \leq \frac{2^{k-1}}{3^k} \). The series \( g = \sum_k g_k \) converges uniformly (Weierstrass M-test), defines a continuous extension with \( \|g\|_\infty \leq \sum \frac{2^{k-1}}{3^k} = 1 \). \( \square \)

Infinite Products and Tychonoff’s Theorem

\[ \left\{ \prod_{k \in K} U_k \;\Big|\; U_k \subseteq X_k \text{ open}, \; U_k = X_k \text{ for all but finitely many } k \right\}. \]

The coarser product topology differs from the finer box topology (which allows arbitrary open \( U_k \) at every index) when \( K \) is infinite.

Theorem 4.29. A function \( f : A \to \prod_{k \in K} X_k \) (with the product topology) is continuous iff each component \( f_k = p_k \circ f : A \to X_k \) is continuous, where \( p_k \) is the projection.

Theorem 4.31 (Tychonoff’s Theorem). The product of any indexed family of compact spaces is compact in the product topology.

Proof sketch. Using Zorn’s Lemma, extend any collection \( \mathcal{T} \) of closed sets with the finite intersection property to a maximal such collection \( \mathcal{S} \) (closed under finite intersections). For each \( k \), the collection \( \{p_k(A) \mid A \in \mathcal{S}\} \) of projected sets still has the finite intersection property in the compact space \( X_k \), so choose \( a_k \in \bigcap_{A \in \mathcal{S}} \overline{p_k(A)} \). The point \( a = (a_k) \) lies in every \( A \in \mathcal{S} \) (because every basic open neighbourhood of \( a \) meets every \( A \in \mathcal{S} \), hence in every element of \( \mathcal{T} \). \( \square \)

Nets

Definition 4.32. A directed set is a set \( K \) with a binary relation \( \leq \) that is reflexive, transitive, and directed (for all \( a, b \) there exists \( c \) with \( a \leq c \) and \( b \leq c \). A net in a topological space \( X \) is an indexed family \( (x_k)_{k \in K} \) where \( K \) is directed. A net converges to \( a \in X \) when for every open \( U \ni a \) there exists \( m \in K \) such that \( k \geq m \implies x_k \in U \).

Theorem 4.34. Let \( X \) be a topological space, \( A \subseteq X \), \( a \in X \). Then \( a \in \overline{A} \) iff there is a net \( (x_k) \) in \( A \) with \( x_k \to a \).

Theorem 4.35. Let \( f : A \subseteq X \to Y \). Then \( f \) is continuous on \( A \) iff for every net \( (x_k) \) in \( A \) with \( x_k \to a \in A \), we have \( f(x_k) \to f(a) \) in \( Y \).

Strong and Weak Topologies and the Banach-Alaoglu Theorem

Definition 4.36 (Final/Strong Topology). Given functions \( f_k : X_k \to Y \), the final (strong) topology on \( Y \) is the finest topology making all \( f_k \) continuous.

Definition 4.38 (Initial/Weak Topology). Given functions \( f_k : X \to Y_k \), the initial (weak) topology on \( X \) is the coarsest topology making all \( f_k \) continuous — the topology generated by \( \{f_k^{-1}(U) \mid k \in K, \; U \subseteq Y_k \text{ open}\} \).

Definition 4.41 (Weak and Weak\( ^* \) Topologies). Let \( U \) be a normed space.

• The weak topology on \( U \) is the initial topology with respect to \( (f)_{f \in U^*} \).
• The weak\( ^* \) topology on \( U^* \) is the initial topology with respect to \( (F_u)_{u \in U} \) where \( F_u(f) = f(u) \).

Theorem 4.42. In a normed space \( U \):

1. \( x_k \to a \) in the weak topology iff \( f(x_k) \to f(a) \) for all \( f \in U^* \).
2. \( f_k \to g \) in \( U^* \) with the weak\( ^* \) topology iff \( f_k(x) \to g(x) \) for all \( x \in U \).

Remark 4.43. When \( U \) is infinite-dimensional, the closed unit ball \( B_{U^*}(0,1) \) is not compact in the norm topology on \( U^* \) (by Riesz’s Theorem 3.8). The Banach-Alaoglu Theorem shows it is compact in a weaker topology.

Theorem 4.44 (Banach-Alaoglu Theorem). For any normed space \( U \), the closed unit ball \( B_{U^*}(0,1) = \{f \in U^* \mid \|f\| \leq 1\} \) is compact in the weak\( ^* \) topology.

Proof. Let \( B = \{x \in U \mid \|x\| \leq 1\} \), \( D = \{t \in \mathbb{F} \mid |t| \leq 1\} \), and \( P = D^B = \prod_{u \in B} D \) with the product topology. The restriction map \( R : B_{U^*}(0,1) \to P \), \( R(f)(x) = f(x) \), is injective and continuous (each component \( R_u(f) = f(u) \) is continuous in the weak\( ^* \) topology). The image \( R(B_{U^*}(0,1)) \) is closed in \( P \) (limit points of locally-linear functions on \( B \) extend to linear maps on \( U \). Since \( D \) is compact, \( P \) is compact by Tychonoff. As a closed subset of a compact space, \( R(B_{U^*}(0,1)) \) is compact. Since \( R \) is a homeomorphism onto its image (its inverse is also continuous), \( B_{U^*}(0,1) \) is compact. \( \square \)

Locally Convex Topological Vector Spaces

Definition 4.45. A topological vector space over \( \mathbb{F} \) is a Hausdorff vector space with a topology making addition and scalar multiplication continuous. It is locally convex when its topology has a basis of convex sets.

Example 4.46. For a normed space \( U \): the norm topology, the weak topology \( (U, \text{wk}) \), and the weak\( ^* \) topology \( (U^*, \text{wk}^*) \) are all locally convex topological vector spaces. In particular, the weak topology is Hausdorff (by the Hahn-Banach Theorem, distinct points are separated by functionals) and has a basis of convex sets (finite intersections of sets of the form \( f^{-1}(V) \).

Note 4.49. In a real topological vector space \( U \), the interior of any set is contained in its core: \( A^\circ \subseteq \operatorname{Core}(A) \).

Theorem 4.50 (Hahn-Banach Separation for Topological Vector Spaces). Let \( U \) be a real topological vector space and \( A, B \subseteq U \) disjoint nonempty convex subsets.

1. If \( A \) is open, there exists \( 0 \neq f \in U^* \) and \( c \in \mathbb{R} \) with \( f(x) < c \leq f(y) \) for all \( x \in A \), \( y \in B \).
2. If \( U \) is locally convex, \( A \) is compact, and \( B \) is closed, there exists \( 0 \neq f \in U^* \) and \( c \in \mathbb{R} \) with \( f(x) < c < f(y) \) for all \( x \in A \), \( y \in B \).

This geometric version of the Hahn-Banach Theorem provides the foundation for convex analysis and duality theory in functional analysis. Part (1) follows from Theorem 3.20 combined with Note 4.49 (which ensures the open set \( A \) is contained in its core). Part (2) uses local convexity to thicken \( A \) slightly and separate it from \( B \) with strict inequalities on both sides.