Surjection of Fréchet spaces

The theorem on the surjection of Fréchet spaces is an important theorem, due to Stefan Banach, 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.

Preliminaries

Let X and Y be topological vector spaces and $L:X\to Y$ be a continuous linear map. We use the following notation and definitions:

• The continuous dual space of X is denoted by $X^{*}$.
• The transpose of L is the map ${}^{t}L:Y^{*}\to X^{*}$ defined by $L\left(y^{\prime }\right):=y^{\prime }\circ L$.
  • Note that if $L:X\to Y$ is surjective then ${}^{t}L:Y^{*}\to X^{*}$ will be injective, but the converse is not true in general.
• The weak topology on $X$ (resp. $X^{*}$) is denoted by $\sigma \left(X,X^{*}\right)$ (resp. $\sigma \left(X^{*},X\right)$). The set X endowed with this topology is denoted by $\left(X,\sigma \left(X,X^{*}\right)\right)$.
  • $\sigma \left(X,X^{*}\right)$ is the weakest topology on X making all linear functionals in $X^{*}$ continuous.
• If S is a subset of Y then the polar of S in Y is denoted by $S^{\circ }$.
• If $p:X\to \mathbb {R}$ is a seminorm on X, then $X_{p}$ will denoted the vector space X endowed with the weakest TVS topology making p continuous. A neighborhood basis of $X_{p}$ at the origin consists of the sets $\left\{x\in X:p(x)<r\right\}$ as r ranges over the positive reals. Note that if p is not a norm then $X_{p}$ is not Hausdorff and $\operatorname {Ker} p:=\left\{x\in X:p(x)=0\right\}$ is a linear subspace of X.
  • Note that if p is continuous then the identity map $\operatorname {Id} :X\to X_{p}$ is continuous so we may identify the continuous dual space of $X_{p}$, $X_{p}^{*}$, as a subset of $X^{*}$ via the transpose of the identity map ${}^{t}\operatorname {Id} :X_{p}^{*}\to X^{*}$, which is clearly injective.

Surjection of Fréchet spaces

Theorem (Banach): Let X and Y be Fréchet spaces and let L : X → Y be a continuous linear map. Then L : X → Y is surjective if and only if the following two conditions hold:

1. ${}^{t}L:Y^{*}\to X^{*}$ is injective;
2. the image of ${}^{t}L$, denoted by $\operatorname {Im} {}^{t}L$, is weakly closed in $X^{*}$ (i.e. closed when $X^{*}$ is endowed with the weak-* topology).

Extensions of the theorem

Theorem: Let X and Y be Fréchet spaces and let $L:X\to Y$ be a continuous linear map. Then the following are equivalent:

• $L:X\to Y$ is surjective;
• the following two conditions hold:

1. ${}^{t}L:Y^{*}\to X^{*}$ is injective;
2. the image of ${}^{t}L$, $\operatorname {Im} {}^{t}L$, is weakly closed in $X^{*}$.

• For every continuous seminorm p on X there exists a continuous seminorm q on Y such that the following are true:

1. for every $y\in Y$ there exists some $x\in X$ such that $q\left(L(x)-y\right)=0$;
2. for every $y^{\prime }\in Y$, if ${}^{t}L\left(y^{\prime }\right)\in X_{p}^{*}$ then $y^{\prime }\in Y_{q}^{*}$.

• For every continuous seminorm p on X there exists a linear subspace N of Y such that the following are true:

1. for every $y\in Y$ there exists some $x\in X$ such that $L(x)-y\in N$;
2. for every $y^{\prime }\in Y^{*}$, if ${}^{t}L\left(y^{\prime }\right)\in X_{p}^{*}$ then $y^{\prime }\in N^{\circ }$.

• There is a non-increasing sequence $N_{1}\supseteq N_{2}\supseteq N_{3}\supseteq \cdots$ of closed linear subspaces of Y whose intersection is equal to $\{0\}$ and such that the following are true:

1. For every $y\in Y$ and every positive integer k, there exists some $x\in X$ such that $L(x)-y\in N_{k}$;
2. For every continuous seminorm p on X there exists an integer k such that any $x\in X$ that satisfies $L(x)\in N_{k}$ is the limit, in the sense of the seminorm p, of a sequence $x_{1},x_{2},\ldots$ in elements of X such that $L(x_{i})=0$ for all i.

Lemmas

The following lemmas are used to prove the theorems on the surjectivity of Fréchet spaces. They are useful even on their own.

Theorem: Let X be a Fréchet space and Z be a linear subspace of $X^{*}$. The following are equivalent:

1. Z is weakly closed in $X^{*}$;
2. There exists a basis ${\mathcal {B}}$ of neighborhoods of the origin of X such that for every $B\in {\mathcal {B}}$, $B^{\circ }\cap Z$ is weakly closed;
3. The intersection of Z with every equicontinuous subset E of $X^{*}$ is relatively closed in E (where $X^{*}$ is given the weak topology induced by X and E is given the subspace topology induced by $X^{*}$).

Theorem: On the dual $X^{*}$ of a Fréchet space X, the topology of uniform convergence on compact convex subsets of X is identical to the topology of uniform convergence on compact subsets of X.

Lemma: Let $L:X\to Y$ be a linear map between Hausdorff locally convex TVSs, with X also metrizable. If the map $L:\left(X,\sigma \left(X,X^{*}\right)\right)\to \left(Y,\sigma \left(Y,Y^{*}\right)\right)$ is continuous then $L:X\to Y$ is continuous (where X and Y carry their original topologies).

Applications

Borel's theorem on power series expansions

Theorem (E. Borel): Fix a positive integer n. If P is an arbitrary formal power series in n indeterminants with complex coefficients then there exists a ${\mathcal {C}}^{\infty }$ function $f:\mathbb {R} ^{n}\to \mathbb {C}$ whose Taylor expansion at the origin is identical to P.

That is, suppose that for every n-tuple of non-negative integers $p=\left(p_{1},\ldots ,p_{n}\right)$ we are given a complex number $a_{p}$ (with no restrictions). Then there exists a ${\mathcal {C}}^{\infty }$ function $f:\mathbb {R} ^{n}\to \mathbb {C}$ such that $a_{p}={\frac {\partial f}{\partial x}}{\Bigg \vert }_{x=0}$ for every n-tuple p of non-negative integers.

Linear partial differential operators

Theorem: Let D be a linear partial differential operator with ${\mathcal {C}}^{\infty }$ coefficients in an open subset $U\subseteq \mathbb {R} ^{n}$. The following are equivalent:

1. For every $f\in {\mathcal {C}}^{\infty }(U)$ there exists some $u\in {\mathcal {C}}^{\infty }(U)$ such that $Du=f$.
2. U is D-convex and D is semiglobally solvable.

Here,

• D being semiglobally solvable in U means that for every relatively compact open subset V of U, the following condition holds:

To every $f\in {\mathcal {C}}^{\infty }(U)$ there is some $g\in {\mathcal {C}}^{\infty }(U)$ such that $Dg=f$ in V.

• U being D-convex means that for every compact subsets $K\subseteq U$ and every integer $n\geq 0$, there is a compact subset $C_{n}$ of U such that for every distribution d with compact support in U, the following condition holds:

If ${}^{t}Dd$ is of order $\leq n$ and if $\operatorname {supp} {}^{t}Dd\subseteq K$, then $\operatorname {supp} d\subseteq C_{n}$.

See also

• Epimorphism

References

• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.