Surjection of Fréchet spaces

Let $L\; :\; X\; \backslash to\; Y$ be a continuous linear map between topological vector spaces.

The continuous dual space of $X$ is denoted by $X^\{\backslash prime\}.$

The transpose of $L$ is the map $\{\}^t\; L\; :\; Y^\{\backslash prime\}\; \backslash to\; X^\{\backslash prime\}$ defined by $L\; \backslash left(y^\{\backslash prime\}\backslash right)\; :=\; y^\{\backslash prime\}\; \backslash circ\; L.$ If $L\; :\; X\; \backslash to\; Y$ is surjective then $\{\}^t\; L\; :\; Y^\{\backslash prime\}\; \backslash to\; X^\{\backslash prime\}$ will be injective, but the converse is not true in general.

The weak topology on $X$ (resp. $X^\{\backslash prime\}$) is denoted by $\backslash sigma\backslash left(X,\; X^\{\backslash prime\}\backslash right)$ (resp. $\backslash sigma\backslash left(X^\{\backslash prime\},\; X\backslash right)$). The set $X$ endowed with this topology is denoted by $\backslash left(X,\; \backslash sigma\backslash left(X,\; X^\{\backslash prime\}\backslash right)\backslash right).$ The topology $\backslash sigma\backslash left(X,\; X^\{\backslash prime\}\backslash right)$ is the weakest topology on $X$ making all linear functionals in $X^\{\backslash prime\}$ continuous.

If $S\; \backslash subseteq\; Y$ then the polar of $S$ in $Y$ is denoted by $S^\{\backslash circ\}.$

If $p\; :\; X\; \backslash to\; \backslash R$ is a seminorm on $X$, then $X\_p$ will denoted the vector space $X$ endowed with the weakest TVS topology making $p$ continuous.{{sfn|Trèves|2006|pp=378-384}} A neighborhood basis of $X\_p$ at the origin consists of the sets as $r$ ranges over the positive reals. If $p$ is not a norm then $X\_p$ is not Hausdorff and $\backslash ker\; p\; :=\; \backslash left\backslash \{\; x\; \backslash in\; X\; :\; p(x)\; =\; 0\; \backslash right\backslash \}$ is a linear subspace of $X$.

If $p$ is continuous then the identity map $\backslash operatorname\{Id\}\; :\; X\; \backslash to\; X\_p$ is continuous so we may identify the continuous dual space $X\_p^\{\backslash prime\}$ of $X\_p$ as a subset of $X^\{\backslash prime\}$ via the transpose of the identity map $\{\}^\{t\}\; \backslash operatorname\{Id\}\; :\; X\_p^\{\backslash prime\}\; \backslash to\; X^\{\backslash prime\},$ which is injective.

{{math theorem|name=Theorem{{sfn|Trèves|2006|pp=378-384}}|note=Banach|math_statement=If $L\; :\; X\; \backslash to\; Y$ is a continuous linear map between two Fréchet spaces, then $L\; :\; X\; \backslash to\; Y$ is surjective if and only if the following two conditions both hold:

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

}}

{{math theorem|name=Theorem{{sfn|Trèves|2006|pp=378-384}}|note=|style=|math_statement=

If $L\; :\; X\; \backslash to\; Y$ is a continuous linear map between two Fréchet spaces then the following are equivalent:

1. $L\; :\; X\; \backslash to\; Y$ is surjective.

2. The following two conditions hold:

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

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

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

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

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

5. There is a non-increasing sequence $N\_1\; \backslash supseteq\; N\_2\; \backslash supseteq\; N\_3\; \backslash supseteq\; \backslash cdots$ of closed linear subspaces of $Y$ whose intersection is equal to $\backslash \{\; 0\; \backslash \}$ and such that the following are true:

   1. for every $y\; \backslash in\; Y$ and every positive integer $k$, there exists some $x\; \backslash in\; X$ such that $L(x)\; -\; y\; \backslash in\; N\_k$;
   2. for every continuous seminorm $p$ on $X$ there exists an integer $k$ such that any $x\; \backslash in\; X$ that satisfies $L(x)\; \backslash in\; N\_k$ is the limit, in the sense of the seminorm $p$, of a sequence $x\_1,\; x\_2,\; \backslash ldots$ in elements of $X$ such that $L\backslash left(x\_i\backslash right)\; =\; 0$ for all $i.$

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

{{math theorem|name=Theorem{{sfn|Trèves|2006|pp=378-384}} |note=|style=|math_statement=

Let $X$ be a Fréchet space and $Z$ be a linear subspace of $X^\{\backslash prime\}.$

The following are equivalent:

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

}}

{{math theorem|name=Theorem{{sfn|Trèves|2006|pp=378-384}} |note=|style=|math_statement=

On the dual $X^\{\backslash prime\}$ 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 $X$.

}}

{{math theorem|name=Theorem{{sfn|Trèves|2006|pp=378-384}}|note=|style=|math_statement=

Let $L\; :\; X\; \backslash to\; Y$ be a linear map between Hausdorff locally convex TVSs, with $X$ also metrizable.

If the map $L\; :\; \backslash left(X,\; \backslash sigma\backslash left(X,\; X^\{\backslash prime\}\backslash right)\backslash right)\; \backslash to\; \backslash left(Y,\; \backslash sigma\backslash left(Y,\; Y^\{\backslash prime\}\backslash right)\backslash right)$ is continuous then $L\; :\; X\; \backslash to\; Y$ is continuous (where $X$ and $Y$ carry their original topologies).

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{{math theorem|name=Theorem{{sfn|Trèves|2006|p=390}} |note=E. Borel|style=|math_statement=

Fix a positive integer $n$.

If $P$ is an arbitrary formal power series in $n$ indeterminates with complex coefficients then there exists a $\backslash mathcal\{C\}^\{\backslash infty\}$ function $f\; :\; \backslash R^n\; \backslash to\; \backslash Complex$ whose Taylor expansion at the origin is identical to $P$.

That is, suppose that for every $n$-tuple of non-negative integers $p\; =\; \backslash left(p\_1,\; \backslash ldots,\; p\_n\backslash right)$ we are given a complex number $a\_p$ (with no restrictions). Then there exists a $\backslash mathcal\{C\}^\{\backslash infty\}$ function $f\; :\; \backslash R^n\; \backslash to\; \backslash Complex$ such that $a\_p\; =\; \backslash left(\backslash partial\; /\; \backslash partial\; x\backslash right)^p\; f\; \backslash bigg\backslash vert\_\{x\; =\; 0\}$ for every $n$-tuple $p.$

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{{See also|Distribution (mathematics)}}

{{math theorem|name=Theorem{{sfn|Trèves|2006|p=392}}|note=|style=|math_statement=

Let $D$ be a linear partial differential operator with $\backslash mathcal\{C\}^\{\backslash infty\}$ coefficients in an open subset $U\; \backslash subseteq\; \backslash R^n.$

The following are equivalent:

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

}}

$D$ being {{em|semiglobally solvable in $U$}} means that for every relatively compact open subset $V$ of $U$, the following condition holds:

:to every $f\; \backslash in\; \backslash mathcal\{C\}^\{\backslash infty\}(U)$ there is some $g\; \backslash in\; \backslash mathcal\{C\}^\{\backslash infty\}(U)$ such that $D\; g\; =\; f$ in $V$.

$U$ being {{em|$D$-convex}} means that for every compact subset $K\; \backslash subseteq\; U$ and every integer $n\; \backslash geq\; 0,$ 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 $\{\}^\{t\}\; D\; d$ is of order $\backslash leq\; n$ and if $\backslash operatorname\{supp\}\; \{\}^\{t\}\; D\; d\; \backslash subseteq\; K,$ then $\backslash operatorname\{supp\}\; d\; \backslash subseteq\; C\_n.$

• {{annotated link|Epimorphism}}
• {{annotated link|Exponential formula}}
• {{annotated link|Open mapping theorem (functional analysis)}}

{{reflist|group=note}}

{{reflist}}

• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}
• {{Trèves François Topological vector spaces, distributions and kernels}}

{{Functional analysis}}

Category:Theorems in functional analysis