In the best-approximation problem of seperation theorems in convex analysis, there is the notion of a "strong norm", in the sense that
If $\| x^1 + x^2 \| = \| x^1 \| + \|x^2 \| $, $x_1 , x_2 \neq 0$ $\implies$ $x^1=\lambda x^2$ for some $\lambda>0$.
How is the connection to norms that are induced from scalar products on unitary/pre-Hilbert spaces (i.e. $\| x \| = \sqrt{ \langle x ,x \rangle }$) as they are studied in functional analysis. Are these induced norms always "strong"?