Normalization in $L^{p}$ and $L^{q}$

Question

Given a function $f$ in $L^{p}\cap L^{q}$ where $0<p,q<\infty$, can $f$ always be normalized such that $\| f \|_p=\| f \|_q=1$?

Answer 1

No.

For example take your measure space to be $[0,1]$ with Lebesgue measure. Take $f(x)=2x$, $p=1,q=2$. Then $||f||_p=1$. To normalize $f$ you can only multiply it by a constant- and that would disturb the $||.||_p$.

Answer 2

For an even simpler counterexample, let $f(x)\equiv 0$.