Surjection of Fréchet spaces

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

Theorem[1] — 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. ${\displaystyle {}^{t}L:Y^{\prime }\to X^{\prime }}$ is injective;
   2. the image ${\displaystyle \operatorname {Im} {}^{t}L}$ of ${\displaystyle {}^{t}L}$ is weakly closed in ${\displaystyle X^{\prime }.}$
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 ${\displaystyle y^{\prime }\in Y,}$ if ${\displaystyle {}^{t}L\left(y^{\prime }\right)\in X_{p}^{\prime }}$ then ${\displaystyle y^{\prime }\in Y_{q}^{\prime }.}$
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 ${\displaystyle y^{\prime }\in Y^{\prime },}$ if ${\displaystyle {}^{t}L\left(y^{\prime }\right)\in X_{p}^{\prime }}$ then ${\displaystyle y^{\prime }\in N^{\circ }.}$
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 ${\displaystyle x_{1},x_{2},\ldots }$ in elements of ${\displaystyle X}$ such that ${\displaystyle L\left(x_{i}\right)=0}$ for all ${\displaystyle i.}$

Theorem[1] — Let ${\displaystyle X}$ be a Fréchet space and ${\displaystyle Z}$ be a linear subspace of ${\displaystyle X^{\prime }.}$ The following are equivalent:

1. ${\displaystyle Z}$ is weakly closed in ${\displaystyle X^{\prime }}$;
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 ${\displaystyle X^{\prime }}$ is relatively closed in ${\displaystyle E}$ (where ${\displaystyle X^{\prime }}$ is given the weak topology induced by ${\displaystyle X}$ and ${\displaystyle E}$ is given the subspace topology induced by ${\displaystyle X^{\prime }}$).

Theorem[1] — On the dual ${\displaystyle X^{\prime }}$ 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[1] — Let ${\displaystyle L:X\to Y}$ be a linear map between Hausdorff locally convex TVSs, with ${\displaystyle X}$ also metrizable. If the map ${\displaystyle L:\left(X,\sigma \left(X,X^{\prime }\right)\right)\to \left(Y,\sigma \left(Y,Y^{\prime }\right)\right)}$ is continuous then ${\displaystyle L:X\to Y}$ is continuous (where ${\displaystyle X}$ and ${\displaystyle Y}$ carry their original topologies).