This is a list of miscellaneous theorems from Functional Analysis.
Hahn-Banach: Let be a real-valued function on a normed linear space with
(i) Positive homogeneity
Let denote a linear subspace of on which for all . Then can be extended to all of : for all .
Geometric Hahn-Banach / Hyperplane Separation Theorem: Let be a nonempty convex subset, . Let be a point outside . Then there exists a linear functional (depending on ) such that for all ; .
Riesz Representation Theorem: Let be a linear functional on a Hilbert space that is bounded: . Then for some unique .
Principle of Uniform Boundedness: A weakly convergent sequence is uniformly bounded in norm.
Open Mapping Principle/Theorem: Let be Banach spaces. Let be a bounded linear map onto all of . Then maps open sets onto open sets.
Closed Graph Theorem: Let be Banach spaces, be a closed linear map. Then is continuous.