I've seen in lecture notes that norm on any Hilbert space is strictly convex means
"$\|x\|=\|y\|=1, \quad\|x+y\|=2 \Rightarrow x=y$"
But why this means strict convexity? I thought strict convexity means $\forall x_{1} \neq x_{2} \in H, \forall t \in(0,1) : \quad \|t x_{1}+(1-t) x_{2}\|<t \|f\left(x_{1}\right)\|+(1-t)\| f\left(x_{2}\right)\|$?
It would be best if you can please give a complete proof of why norm on "any" Hilbert space is strictly convex