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

Question

Is $C([0,1])$ equipped with the supremum-norm strictly convex?

With strict convexity given by: if $f\neq g$ and $\Vert f\Vert=1=\Vert g\Vert$ then $\Vert f+g\Vert<2$.

I guess it is not, but how can I prove it?

Answer 1

With the strict inequality, it is false.

Take $f,g\colon[0,1]\to[0,1]$ defined by $f(x)=1$ (constant) and $g(x)=x$. Then $\lVert f\rVert_\infty=\lVert g\rVert_\infty=1$, $\lVert f+g\rVert_\infty=f(1)+g(1)=2$, and $f\neq g$.

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With the non-strict inequality (your previous question, before the edit), it is an immediate consequence of the triangle inequality.