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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