Corollary 1

Tag 07A9, permanent. Cite it as [Bourbaki, Tag 07A9].

Topological Vector Spaces, chapter I, § 3. Metrisable topological vector spaces

If E and F are two complete metrisable vector spaces over a non-discrete valued division ring, then every bijective continuous linear mapping of E on F is an isomorphism.

In particular, if E and F are two complete normed spaces, there exists a number $a > 0$ such that $\|u(x)\| \geq a.\|x\|$ for all $x \in E$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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