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. ${\displaystyle {}^{t}L:Y^{*}\to X^{*}}$ is injective, and
2. the image of ${\displaystyle {}^{t}L,}$ denoted by ${\displaystyle \operatorname {Im} {}^{t}L,}$ is weakly closed in ${\displaystyle X^{*}}$ (i.e. closed when ${\displaystyle X^{*}}$ 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. ${\displaystyle {}^{t}L:Y^{*}\to X^{*}}$ is injective;
   2. the image of ${\displaystyle {}^{t}L,}$ ${\displaystyle \operatorname {Im} {}^{t}L,}$ is weakly closed in ${\displaystyle X^{*}.}$
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 ${\displaystyle y\in Y}$ there exists some ${\displaystyle x\in X}$ such that ${\displaystyle q\left(L(x)-y\right)=0}$;
   2. for every ${\displaystyle y^{\prime }\in Y,}$ if ${\displaystyle {}^{t}L\left(y^{\prime }\right)\in X_{p}^{*}}$ then ${\displaystyle y^{\prime }\in Y_{q}^{*}.}$
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 ${\displaystyle y\in Y}$ there exists some ${\displaystyle x\in X}$ such that ${\displaystyle L(x)-y\in N}$;
   2. for every ${\displaystyle y^{\prime }\in Y^{*},}$ if ${\displaystyle {}^{t}L\left(y^{\prime }\right)\in X_{p}^{*}}$ 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 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 k, there exists some ${\displaystyle x\in X}$ such that ${\displaystyle L(x)-y\in N_{k}}$;
   2. For every continuous seminorm p on X there exists an integer 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 p, of a sequence ${\displaystyle x_{1},x_{2},\ldots }$ in elements of X such that ${\displaystyle L(x_{i})=0}$ for all i.

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

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

Theorem[1] — On the dual ${\displaystyle X^{*}}$ 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.

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