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.