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My question is quite simple:

Is the space $C([0,1])$ strictly convex?

Where strict convexity is defined as: if $\Vert y+x \Vert=\Vert y\Vert+\Vert x\Vert$ then it implies that is exists an $\lambda>0$ s.t. $x=\lambda y$.

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No. Consider $y(t) = t^m$ and $x(t) = t^n$ for some $n\neq m$. Then

$$2 = \| x + y\| = \|x \| + \|y\| $$

but $x\neq \lambda y$.

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