Surjection of Fréchet spaces

The theorem on the surjection of Fréchet spaces is an important theorem, due to Stefan Banach, that characterizes when a continuous linear operator between Fréchet spaces is surjective.

The importance of this theorem is related to the open mapping theorem, which states that a continuous linear surjection between Fréchet spaces is an open map. Often in practice, one knows that they have a continuous linear map between Fréchet spaces and wishes to show that it is surjective in order to use the open mapping theorem to deduce that it is also an open mapping. This theorem may help reach that goal.

Preliminaries, definitions, and notation
Let $$L : X \to Y$$ be a continuous linear map between topological vector spaces.

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

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

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

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

If $$p : X \to \R$$ is a seminorm on $$X$$, then $$X_p$$ will denoted the vector space $$X$$ endowed with the weakest TVS topology making $$p$$ continuous. A neighborhood basis of $$X_p$$ at the origin consists of the sets $$\left\{ x \in X : p(x) < r \right\}$$ as $$r$$ ranges over the positive reals. If $$p$$ is not a norm then $$X_p$$ is not Hausdorff and $$\ker p := \left\{ x \in X : p(x) = 0 \right\}$$ is a linear subspace of $$X$$. If $$p$$ is continuous then the identity map $$\operatorname{Id} : X \to X_p$$ is continuous so we may identify the continuous dual space $$X_p^{\prime}$$ of $$X_p$$ as a subset of $$X^{\prime}$$ via the transpose of the identity map $${}^{t} \operatorname{Id} : X_p^{\prime} \to X^{\prime},$$ which is injective.

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

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Borel's theorem on power series expansions
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Linear partial differential operators
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$$D$$ being means that for every relatively compact open subset $$V$$ of $$U$$, the following condition holds:


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

$$U$$ being means that for every compact subset $$K \subseteq U$$ and every integer $$n \geq 0,$$ 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 $$\leq n$$ and if $$\operatorname{supp} {}^{t} D d \subseteq K,$$ then $$\operatorname{supp} d \subseteq C_n.$$