Let $(X,\mathcal{M},\mu)$ be a measure space and $1\leq p\leq\infty$. Suppose $f,g\in L^p(X,\mathcal{M},\mu)$ with $f(x)g(x)=0$ for almost all $x\in X$ and $\Vert f\Vert_p=\Vert g \Vert_p=1$. How to verify by direct calculation that
$$\Vert f+g\Vert_p^2+\Vert f-g\Vert_p^2=2(\Vert f\Vert_p^2+\Vert g\Vert_p^2)$$ if and ony if $p=2$?
Let us first consider the norms $\|f+g\|_p$ and $\|f-g\|_p$ for $p\in[1,\infty)$:
$$\|f\pm g\|^p_p=\int_X|f\pm g|^pd\mu=\int_{[fg=0]}|f\pm g|^pd\mu=\int_{[f\neq0]}|f\pm g|^pd\mu+\int_{[g\neq0]}|f\pm g|^pd\mu=$$ $$=\int_{[f\neq0]}|f|^pd\mu+\int_{[g\neq0]}|g|^pd\mu=\int_X|f|^pd\mu+\int_X|g|^pd\mu=\|f\|^p_p+\|g\|^p_p=1+1=2$$
where the third and fourth transition follow from the fact that $fg=0$ (a.e.). Now from the parallelogram law we get that $2^{2/p}+2^{2/p}=4$, which is true if and only if $p=2$.
The case of $p=\infty$ is similar, just use the supremum norm.