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
(ii) Subadditivity
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.