Functional Analysis List of Theorems

This is a list of miscellaneous theorems from Functional Analysis.

Hahn-Banach: Let $p$ be a real-valued function on a normed linear space $X$ with

(i) Positive homogeneity

(ii) Subadditivity

Let $Y$ denote a linear subspace of $X$ on which $l(y)\leq p(y)$ for all $y\in Y$ . Then $l$ can be extended to all of $X$ : $l(x)\leq p(x)$ for all $x\in X$ .

Geometric Hahn-Banach / Hyperplane Separation Theorem: Let $K$ be a nonempty convex subset, $K=int(K)$ . Let $y$ be a point outside $K$ . Then there exists a linear functional $l$ (depending on $y$ ) such that $l(x)<c$ for all $x\in K$ ; $l(y)=c$ .

Riesz Representation Theorem: Let $l(x)$ be a linear functional on a Hilbert space $H$ that is bounded: $|l(x)|\leq c\|x\|$ . Then $l(x)=\langle x,y\rangle$ for some unique $y\in H$ .

Principle of Uniform Boundedness: A weakly convergent sequence $\{x_n\}$ is uniformly bounded in norm.

Open Mapping Principle/Theorem: Let $X, U$ be Banach spaces. Let $M:X\to U$ be a bounded linear map onto all of $U$ . Then $M$ maps open sets onto open sets.

Closed Graph Theorem: Let $X,U$ be Banach spaces, $M:X\to U$ be a closed linear map. Then $M$ is continuous.

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