Preliminaries, definitions, and notation

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. ${\displaystyle {}^{t}L:Y^{\mathrm{\prime }}\to X^{\mathrm{\prime }}}$ isinjective, and
2. theimage 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^[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. ${\displaystyle {}^{t}L:Y^{\mathrm{\prime }}\to X^{\mathrm{\prime }}}$ isinjective;
   2. theimage${\displaystyle \mathrm{Im}{}^{t}L}$ 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 anon-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

Theorem^[1]—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^[1]—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 ofuniform convergence on compact subsets of${\displaystyle X}$.

Theorem^[1]—Let${\displaystyle L:X\to Y}$ be a linear map between Hausdorfflocally convex TVSs, with${\displaystyle X}$ alsometrizable. 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^[2] (E. Borel)—Fix a positive integer${\displaystyle n}$. If${\displaystyle P}$ is an arbitraryformal 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^[3]—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.

See also

References

1. ^^abcdefgTrèves 2006, pp. 378–384.
2. ^Trèves 2006, p. 390.
3. ^Trèves 2006, p. 392.

Bibliography