Showing that $L^{\infty}([0,1])$ is not strictly convex

Question

Can somebody give an example that shows that $L^{\infty}([0,1])$ (regarding $|| \cdot ||_{\infty}$) is not strictly convex?

Thanks in advance!

Answer 1

Let $\chi_{(a,b)} $be a characteristic function of an open interval $(a,b) .$ Take $f=\chi_{\left(0,\frac{1}{2}\right)} , g =\chi_{\left(0,\frac{1}{4}\right)} .$ Then $$||f||=||g|| =\frac{1}{2} ||f+g|| =1$$ but $$f\neq g.$$