Hello,

I am considering the banach-space $\displaystyle L^{p}(\mathbb{R})$ and the functions:

$\displaystyle f=\chi_{[0,1[}$

$\displaystyle g=\chi_{[1,2[}$

I have calculated the numbers $\displaystyle \Vert f \Vert_{p}$, $\displaystyle \Vert g \Vert_{p}$, $\displaystyle \Vert f+g \Vert_{p}$ and $\displaystyle \Vert f-g \Vert_{p}$ (they all have value 1).

How do I, using the above numbers, conclude that $\displaystyle L^{p}(\mathbb{R})$ does not form a hilbertspace for $\displaystyle p \neq 2$ ?

Thanks.