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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"?

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Let $\| \cdot \|$ be an induced norm. Then for all x,y : \begin{align*}\|x+y\|^2 &= \langle x+y,x+y \rangle \\ &= \langle x,x \rangle + 2 \Re \langle x,y \rangle + \langle y,y \rangle \\ &\leq \|x\|^2 + 2 |\langle x,y \rangle | + \|y\|^2 \\ &\leq \|x\|^2+2\|x\| \|y\| + \|y\|^2 = (\|x\|+\|y\|)^2\end{align*} The last $\leq$ is $=$ if and only if $x = \lambda y$ (Cauchy-Schwarz). So if $\|x+y\| = \|x\|+\|y\|$, this implies $x = \lambda y$ for some $\lambda >0$.

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