Equivalent definition of the $L^p$ norm

Question

Is it true for $1 \leq p<\infty$ that for $f \in L^p(\mathbb R)$ and $q$ is such that $1/p+1/q=1$, $$\|f\|_p= \sup \left\{\int_\mathbb R fg : g\in L^q(\mathbb R), \|g\|_q \leq 1 \right\} ?$$

The integral is with respect to the Lebesgue measure.

Answer 1

By Hölder's inequality, $LHS \geqslant RHS$, and for the choice $g:=f^{\frac qp}$, assuming without loss of generality that $f\geqslant 0$, we can see that $LHS\leqslant RHS$.

Answer 2

Let $f = \chi_{[0,2]}$. Then $||f||_2 = \sqrt 2$. On the other hand, if $g = 100 \chi_{[0,2]}$, then $\int fg = 200$, a contradiction.