Corollary 4

Tag 07V0, permanent. Cite it as [Bourbaki, Tag 07V0].

Topological Vector Spaces, chapter IV, § 4. Strict morphisms of Fréchet spaces

— Suppose E and F are Banach spaces. For $^t u$ to be surjective, it is necessary and sufficient that there exist a real number $r > 0$ such that $\| x \| \leq r. \| u(x) \|$ for all $x \in E$ . This is simply a reformulation of the equivalence of the conditions (i) and (iii) of cor. 1.

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

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