The theorem on the surjection of Fréchet spaces is an important theorem, due to Stefan Banach,[1] that characterizes when a continuous linear operator between Fréchet spaces is surjective.
The importance of this theorem is related to the open mapping theorem, which states that a continuous linear surjection between Fréchet spaces is an open map.
Often in practice, one knows that they have a continuous linear map between Fréchet spaces and wishes to show that it is surjective in order to use the open mapping theorem to deduce that it is also an open mapping.
This theorem may help reach that goal.
be a continuous linear map between topological vector spaces.
The continuous dual space of
will be injective, but the converse is not true in general.
endowed with this topology is denoted by
making all linear functionals in
will denoted the vector space
endowed with the weakest TVS topology making
at the origin consists of the sets
ranges over the positive reals.
is continuous then the identity map
via the transpose of the identity map
is a continuous linear map between two Fréchet spaces, then
is surjective if and only if the following two conditions both hold: Theorem[1] — If
is a continuous linear map between two Fréchet spaces then the following are equivalent: The following lemmas are used to prove the theorems on the surjectivity of Fréchet spaces.
The following are equivalent: Theorem[1] — On the dual
, the topology of uniform convergence on compact convex subsets of
is identical to the topology of uniform convergence on compact subsets of
be a linear map between Hausdorff locally convex TVSs, with
Theorem[2] (E. Borel) — Fix a positive integer
is an arbitrary formal power series in
indeterminates with complex coefficients then there exists a
whose Taylor expansion at the origin is identical to
be a linear partial differential operator with
coefficients in an open subset
means that for every relatively compact open subset
-convex means that for every compact subset