The Fundamental Theorems

4.1 Hahn-Banach Theorem

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 $\varphi(x_0) = \|x_0\|$.

4.2 Open Mapping Theorem

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 $T^{-1} \in \mathcal{B}(Y, X)$.

4.3 Closed Graph Theorem

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 $X \times Y$.

4.4 Uniform Boundedness Principle

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 $\sup_\alpha \|T_\alpha\| < \infty$. $\blacksquare$