Corollary

— Suppose E and F are Banach spaces. The following conditions are equivalent : (i) u is surjective. (ii) There exists a real number r > 0, such that $\|f\| \leq r.\|^{t}u(f)\|$ for all $f \in F'$. (iii) For all $y \in F$, we have $\sup_{\substack{f \in F' \\ \|f \circ u\| \leq 1}} |f(y)| < + \infty$.

The conditions (ii) and (iii) are in fact reformulations of conditions (ii) and (iv) of prop. 4 for Banach spaces.