# functional analysis – a Hilbert space is strictly convex

Prove that a Hilbert space is strictly convex in the following sense:

$$U, v ∈ E × E,$$ with $$u ne v, || u || _E = || v || _E = 1$$ and $$∈t ∈]0, 1[[[[$$ we have :

$$| you + (1 – t) v | <1$$.

I've tried to prove that a norm is strictly convex in a Hilbert space but that it could not go far.

Thank you for your help or any reference to a book on this topic.