Suppose $L^p[a,b]$ is the normed space with the usual norm $\|f\|_p=(\int_a^b|f(x)|^p\mathrm{d}x)^{1/p}$. By the parallelogram equality, we know the norm is induced by an inner product iff $p=2$. However, for $p\neq 2$, I wonder can we define an inner product $(\cdot, \cdot)$ on $L^p$, inducing a new norm $\|\cdot\|=\sqrt{(\cdot,\cdot)}$ such that the new norm is equivalent to the usual $L^p$ norm.
I tend to believe this is not true. Appreciate a valid proof!