The theorem on the
surjection
In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
of
Fréchet spaces is an important theorem, due to
Stefan Banach
Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an original ...
, that characterizes when a
continuous linear operator
In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.
An operator between two normed spaces is a bounded linear ...
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
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets.
That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y.
Likewise, ...
. 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.
Preliminaries, definitions, and notation
Let
be a continuous linear map between
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s.
The
continuous dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V,'' together with the vector space structure of pointwise addition and scalar multiplication by const ...
of
is denoted by
The
transpose
In linear algebra, the transpose of a Matrix (mathematics), matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other ...
of
is the map
defined by
If
is surjective then
will be
injective
In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...
, but the converse is not true in general.
The
weak topology on
(resp.
) is denoted by
(resp.
). The set
endowed with this topology is denoted by
The topology
is the weakest topology on
making all linear functionals in
continuous.
If
then the
polar of
in
is denoted by
If
is a
seminorm
In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
on
, then
will denoted the vector space
endowed with the weakest
TVS topology making
continuous. A neighborhood basis of
at the origin consists of the sets
as
ranges over the positive reals. If
is not a norm then
is not
Hausdorff and
is a linear subspace of
.
If
is continuous then the identity map
is continuous so we may identify the continuous dual space
of
as a subset of
via the transpose of the identity map
which is
injective
In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...
.
Surjection of Fréchet spaces
Extensions of the theorem
Lemmas
The following lemmas are used to prove the theorems on the surjectivity of Fréchet spaces. They are useful even on their own.
Applications
Borel's theorem on power series expansions
Linear partial differential operators
being means that for every
relatively compact open subset
of
, the following condition holds:
:to every
there is some
such that
in
.
being means that for every compact subset
and every integer
there is a compact subset
of
such that for every
distribution with compact support in
, the following condition holds:
:if
is of order
and if
then
See also
*
*
*
References
Bibliography
*
*
*
{{Functional analysis
Theorems in functional analysis