Proposition 3

Let E and F be Banach spaces. The following sets are open subsets of the Banach space $\mathscr{L}(E; F):$

a) The set $\mathscr{I}(E; F)$ of the isomorphisms of E onto F ;

b) The set $\mathscr{M} \mathscr{D}(E; F)$ of injective direct morphisms of E into F, and more precisely, for every closed vector subspace $F_1$ of F, the set $\mathscr{M}_{F_1}(E; F)$ of elements of $\mathscr{M} \mathscr{D}(E; F)$ whose image is a topological complement of $F_1$;

c) The set $\mathscr{E} \mathscr{D}(E; F)$ of surjective direct morphisms of E onto F, and more precisely, for every closed vector subspace $E_1$ of E, the set $\mathscr{E}_{E_1}(E; F)$ of elements of $\mathscr{E} \mathscr{D}(E; F)$ whose kernel is a topological complement of $E_1$.

Moreover, the mapping $u\mapsto u^{-1}$ of $\mathscr{I}(E; F)$ onto $\mathscr{I}(F; E)$ is analytic.

By definition, $\mathscr{I}(E; E)$ is the set of invertible elements of the complete normed algebra $\mathscr{L}(E)$. By TG, IX, p. 40, Prop. 14, it is an open subset of $\mathscr{L}(E)$. The mapping $u\mapsto u^{-1}$ of $\mathscr{I}(E; E)$ into $\mathscr{I}(E; E)$ is analytic (LIE, III, § 1, n$^o1$, Example 2).

If the set $\mathscr{I}(E; F)$ is empty, it is open. Otherwise, let $u_0$ be an isomorphism of E onto F. The mapping $v\mapsto u_0\circ v$ is then an isomorphism of $\mathscr{L}(E)$ onto $\mathscr{L}(E; F)$ which transforms $\mathscr{I}(E; E)$ into $\mathscr{I}(E; F)$. Therefore the set $\mathscr{I}(E; F)$ is open in $\mathscr{L}(E; F)$. Moreover, if $u=u_0\circ v$ is an element of $\mathscr{I}(E; F)$, one has $u^{-1}=v^{-1}\circ u^{-1}_0$, and therefore the mapping $u\mapsto u^{-1}$ of $\mathscr{I}(E; F)$ onto $\mathscr{I}(F; E)$ is analytic.

Let $F_1$ be a closed vector subspace of F. For every $u\in \mathscr{L}(E; F)$, let $\overline{u}$ be the element of $\mathscr{L}(E\times F_1; F)$ defined by $\overline{u}(x, y) =u(x) +y$. The mapping $u\mapsto \overline{u}$ of $\mathscr{L}(E; F)$ into $\mathscr{L}(E\times F_1; F)$ is continuous, and $\mathscr{M}_{F_1}(E; F)$ is the set of elements $u$ of $\mathscr{L}(E; F)$ such that $\overline{u}$ belongs to $\mathscr{I}(E\times F_1; F)$. Since $\mathscr{I}(E\times F_1; F)$ is open in $\mathscr{L}(E\times F_1; F)$ by what precedes, the set $\mathscr{M}_{F_1}(E; F)$ is open in $\mathscr{L}(E; F)$. The same is true of $\mathscr{M} \mathscr{D}(E; F)$, which is the union of the $\mathscr{M}_{F_1}(E; F)$ when $F_1$ ranges over the set of closed vector subspaces of F.

Let $E_1$ be a closed vector subspace of E, and let $p$ be the canonical mapping of E onto the quotient Banach space $E/E_1$. In order that an element $u$ of $\mathscr{L}(E; F)$ belong to $\mathscr{E}_{E_1}(E; F)$, it is necessary and sufficient that the mapping $(u, p)$ of E into $F\times E/E_1$ belong to $\mathscr{I}(E; F\times E/E_1)$. As above, it follows that $\mathscr{E}_{E_1}(E; F)$ is open in $\mathscr{L}(E; F)$; the same is true of $\mathscr{E} \mathscr{D}(E; F)$, which is the union of the $\mathscr{E}_{E_1}(E; F)$.