The Fundamental Theorems

Theorem 4.1 (Hahn-Banach, Analytic Form). Let $X$ be a real vector space, $p : X \to \mathbb{R}$ a sublinear functional ($p(x + y) \leq p(x) + p(y)$ and $p(tx) = tp(x)$ for $t \geq 0$), and $f : M \to \mathbb{R}$ a linear functional on a subspace $M \subseteq X$ with $f(x) \leq p(x)$ for all $x \in M$. Then there exists a linear extension $F : X \to \mathbb{R}$ with $F|_M = f$ and $F(x) \leq p(x)$ for all $x \in X$.

Theorem 4.2 (Hahn-Banach, Normed Form). Let $X$ be a normed space and $M \subseteq X$ a subspace. Every bounded linear functional $f \in M^*$ extends to $F \in X^*$ with $\|F\| = \|f\|$.

Corollary 4.3. For every $x_0 \in X$, $x_0 \neq 0$, there exists $\varphi \in X^*$ with $\|\varphi\| = 1$ and $\varphi(x_0) = \|x_0\|$.

Theorem 4.4 (Open Mapping Theorem). If $T \in \mathcal{B}(X, Y)$ is a surjective bounded operator between Banach spaces, then $T$ maps open sets to open sets.

Corollary 4.5 (Bounded Inverse Theorem). If $T \in \mathcal{B}(X, Y)$ is bijective and $X, Y$ are Banach, then $T^{-1} \in \mathcal{B}(Y, X)$.

Theorem 4.6 (Closed Graph Theorem). Let $T : X \to Y$ be a linear operator between Banach spaces. Then $T$ is bounded if and only if its graph $\Gamma(T) = \{(x, Tx) : x \in X\}$ is closed in $X \times Y$.

Theorem 4.7 (Uniform Boundedness Principle / Banach-Steinhaus). Let $X$ be a Banach space and $\{T_\alpha\}_{\alpha \in A} \subseteq \mathcal{B}(X, Y)$ such that $\sup_\alpha \|T_\alpha x\| < \infty$ for each $x \in X$. Then $\sup_\alpha \|T_\alpha\| < \infty$.

Proof sketch. Consider $E = \{x \in X : \sup_\alpha \|T_\alpha x\| \leq n\}$. By the hypothesis, $X = \bigcup_n E_n$. By Baire category, some $E_n$ has nonempty interior. Rescaling shows $\sup_\alpha \|T_\alpha\| < \infty$. $\blacksquare$

Application 1: Extension of linear functionals. The Hahn-Banach theorem guarantees that the dual space $X^*$ is rich enough to separate points: for any $x \neq 0$, there exists $f \in X^*$ with $f(x) \neq 0$.

Application 2: Banach limits. Hahn-Banach can be used to construct a Banach limit: a translation-invariant linear functional $\mathrm{LIM} : \ell^\infty \to \mathbb{R}$ that extends the ordinary limit.

Application 3: Goldstine’s theorem. The unit ball of a Banach space $X$ is weak$*$-dense in the unit ball of the bidual $X^{**}$, a consequence of the Hahn-Banach theorem.

Application 1: Equivalent norms. If $X$ is a Banach space under two norms $\|\cdot\|_1$ and $\|\cdot\|_2$ and there exists $C > 0$ with $\|x\|_1 \leq C\|x\|_2$ for all $x$, then the norms are equivalent.

Application 2: Sum of Banach spaces. If $X$ and $Y$ are closed subspaces of a Banach space $Z$ with $X \cap Y = \{0\}$ and $Z = X + Y$, then the sum is a topological direct sum if and only if both $X$ and $Y$ are closed.

Application. If $T : X \to Y$ is a linear operator between Banach spaces with the property that $x_n \to 0$ and $Tx_n \to y$ implies $y = 0$, then $T$ is bounded.

Application: Hellinger-Toeplitz theorem. If $T : H \to H$ is a linear operator on a Hilbert space satisfying $\langle Tx, y\rangle = \langle x, Ty\rangle$ for all $x, y \in H$, then $T$ is bounded.

Application 1: Fourier series divergence. There exists a continuous function on $[-\pi, \pi]$ whose Fourier series diverges at a point. The UBP shows that the set of partial sum operators $\{S_n\}$ is not pointwise bounded on $C[-\pi, \pi]$.

Application 2: Weak boundedness implies norm boundedness. If a set of operators is bounded in the weak operator topology, it is bounded in the operator norm.

The fundamental theorems of functional analysis all rely on the Baire category theorem:

Theorem 4.8 (Baire Category Theorem). In a complete metric space, the intersection of countably many dense open sets is dense. Equivalently, a complete metric space cannot be expressed as a countable union of nowhere dense sets.

This is the key ingredient in the proofs of the open mapping theorem and the uniform boundedness principle.

Problem 1. Use the Hahn-Banach theorem to show that for any closed subspace $M \subseteq X$, the quotient map $\pi : X \to X/M$ has an isometric right inverse.

Problem 2. Show that if $T : X \to Y$ is a bijective bounded operator between Banach spaces, then $T^{-1}$ is bounded (without using the open mapping theorem, reconstruct the proof).

Problem 3. Let $\{f_n\}$ be a sequence in $X^*$ such that $\lim_{n\to\infty} f_n(x)$ exists for every $x \in X$. Show that the limit functional $f(x) = \lim_n f_n(x)$ is bounded.

Problem 4. Show that the differentiation operator $D : C^1[0, 1] \to C[0, 1]$ defined by $Df = f'$ is unbounded when $C^1[0, 1]$ is given the $\|\cdot\|_\infty$ norm.

The Uniform Boundedness Principle is also called the Banach-Steinhaus theorem or the Resonance theorem. A common formulation is:

Theorem 4.9 (Banach-Steinhaus). If $\{T_n\} \subseteq \mathcal{B}(X, Y)$ and $\lim_{n\to\infty} T_n x$ exists for every $x \in X$, then the limit $Tx = \lim_n T_n x$ defines a bounded linear operator $T$.

Theorem 4.10 (Closed Range Theorem). If $T : X \to Y$ is a bounded linear operator between Banach spaces, then the following are equivalent:

1. The range $R(T)$ is closed in $Y$.
2. The range $R(T^*)$ is closed in $X^*$.
3. $T$ is relatively open: there exists $c > 0$ such that for every $y \in R(T)$, there exists $x$ with $Tx = y$ and $\|x\| \leq c\|y\|$.

Application: Well-posedness of elliptic PDEs. Consider the Dirichlet problem $-\Delta u = f$ on $\Omega$ with $u|_{\partial\Omega} = 0$. The operator $\Delta : H^1_0(\Omega) \to H^{-1}(\Omega)$ is bounded. By the Lax-Milgram lemma (a corollary of Hahn-Banach and Riesz representation), there exists a unique weak solution for every $f$.

Application: Fourier series. The Uniform Boundedness Principle implies that there exist continuous functions whose Fourier series diverge at a point, since the Dirichlet kernels $D_n(t) = \sum_{k=-n}^n e^{ikt}$ have $\|D_n\|_1 \sim \log n \to \infty$ as $n \to \infty$.

Problem 5. Prove that if $X$ is a Banach space and $T : X \to X$ is a bounded linear operator with $\|T\| < 1$, then $I - T$ is invertible (Von Neumann series). Use the open mapping theorem or construct the inverse explicitly.

Problem 6. Show that the closed graph theorem is equivalent to the open mapping theorem (given the bounded inverse theorem).

Problem 7. Let $X$ and $Y$ be Banach spaces and suppose $T : X \to Y$ is a linear operator. Prove that if $x_n \to 0$ and $Tx_n \to y$ implies $y = 0$, then $T$ is bounded.