Surjection of Fréchet spaces

Surjection of Fréchet spaces

Theorem (Banach) — If ${\displaystyle L:X\to Y}$ is a continuous linear map between two Fréchet spaces, then ${\displaystyle L:X\to Y}$ is surjective if and only if the following two conditions both hold:

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

Extensions of the theorem

Theorem — If ${\displaystyle L:X\to Y}$ is a continuous linear map between two Fréchet spaces then the following are equivalent:

1. ${\displaystyle L:X\to Y}$ is surjective.
2. The following two conditions hold:
   1. ${\displaystyle {}^{t}L:Y^{\mathrm{\prime }}\to X^{\mathrm{\prime }}}$ is injective;
   2. the image ${\displaystyle \mathrm{Im}{}^{t}L}$ of ${\displaystyle {}^{t}L}$ is weakly closed in ${\displaystyle X^{\mathrm{\prime }}.}$
3. For every continuous seminorm ${\displaystyle p}$ on ${\displaystyle X}$ there exists a continuous seminorm ${\displaystyle q}$ on ${\displaystyle Y}$ such that the following are true:
   1. for every ${\displaystyle y\in Y}$ there exists some ${\displaystyle x\in X}$ such that ${\displaystyle q(L(x)-y)=0}$;
   2. for every ${\displaystyle y^{\mathrm{\prime }}\in Y,}$ if ${\displaystyle {}^{t}L\left(y^{\mathrm{\prime }}\right)\in X_p^{\mathrm{\prime }}}$ then ${\displaystyle y^{\mathrm{\prime }}\in Y_q^{\mathrm{\prime }}.}$
4. For every continuous seminorm ${\displaystyle p}$ on ${\displaystyle X}$ there exists a linear subspace ${\displaystyle N}$ of ${\displaystyle Y}$ such that the following are true:
   1. for every ${\displaystyle y\in Y}$ there exists some ${\displaystyle x\in X}$ such that ${\displaystyle L(x)-y\in N}$;
   2. for every ${\displaystyle y^{\mathrm{\prime }}\in Y^{\mathrm{\prime }},}$ if ${\displaystyle {}^{t}L\left(y^{\mathrm{\prime }}\right)\in X_p^{\mathrm{\prime }}}$ then ${\displaystyle y^{\mathrm{\prime }}\in N^{\circ }.}$
5. There is a non-increasing sequence ${\displaystyle N_1\supseteq N_2\supseteq N_3\supseteq \cdots }$ of closed linear subspaces of ${\displaystyle Y}$ whose intersection is equal to ${\displaystyle \{0\}}$ and such that the following are true:
   1. for every ${\displaystyle y\in Y}$ and every positive integer ${\displaystyle k}$, there exists some ${\displaystyle x\in X}$ such that ${\displaystyle L(x)-y\in N_k}$;
   2. for every continuous seminorm ${\displaystyle p}$ on ${\displaystyle X}$ there exists an integer ${\displaystyle k}$ such that any ${\displaystyle x\in X}$ that satisfies ${\displaystyle L(x)\in N_k}$ is the limit, in the sense of the seminorm ${\displaystyle p}$, of a sequence ${\displaystyle x_1,x_2,\dots }$ in elements of ${\displaystyle X}$ such that ${\displaystyle L\left(x_i\right)=0}$ for all ${\displaystyle i.}$

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 ${\displaystyle X}$ be a Fréchet space and ${\displaystyle Z}$ be a linear subspace of ${\displaystyle X^{\mathrm{\prime }}.}$ The following are equivalent:

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

Theorem — On the dual ${\displaystyle X^{\mathrm{\prime }}}$ of a Fréchet space ${\displaystyle X}$, the topology of uniform convergence on compact convex subsets of ${\displaystyle X}$ is identical to the topology of uniform convergence on compact subsets of ${\displaystyle X}$.

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

Applications

Borel's theorem on power series expansions

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

That is, suppose that for every ${\displaystyle n}$-tuple of non-negative integers ${\displaystyle p=\left(p_1,\dots ,p_n\right)}$ we are given a complex number ${\displaystyle a_p}$ (with no restrictions). Then there exists a ${\displaystyle {\mathcal{C}}^{\mathrm{\infty }}}$ function ${\displaystyle f:{\mathbb{R}}^n\to \mathbb{C}}$ such that ${\displaystyle a_p={\left(\mathrm{\partial }/\mathrm{\partial }x\right)}^{p}f{|}_{x=0}}$ for every ${\displaystyle n}$-tuple ${\displaystyle p.}$

Linear partial differential operators

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

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

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

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

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

if ${\displaystyle {}^{t}Dd}$ is of order ${\displaystyle \le n}$ and if ${\displaystyle \mathrm{supp}{}^{t}Dd\subseteq K,}$ then ${\displaystyle \mathrm{supp}d\subseteq C_n.}$