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.


About mathtuition88
This entry was posted in math and tagged , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.