adaptation of Hahn-Banach for destination space $\ne \Bbb R$

Question

Let $E$ be a Banach space, $H$ a Hilbert space, $F ⊂ H$ is a closed non-dense subspace, and $ S ∈ \mathcal L(F, E)$. Prove it exists $ T ∈ \mathcal L(H, E)$ s.t. $\|T\| = \|S\|$ and $T (x) = S(x) ∀x ∈ F $.

This would be a corollary of Hahn-Banach theorem if we had $E=\Bbb R$ but I don't see how to adapt the setup of this problem to the analytic version of Hahn-Banach. Or maybe there is other way than Hahn-Banach?

Thank you for your help.

Answer 1

First, you can uniquely extent $T$ to $\bar F$. Then use the orthogonal decomposition of $H = \bar F \oplus F^\perp$ and set $T=0$ on $F^\perp$.