Given any function $0\ne f\in L^p([0,1])$, show that exists a unique function $g\in L^q([0,1])$ where $\|g\|_q=1$ and $$\|f\|_p = \int_0^1 fg\lambda(dx)$$
Showing the existence of such function is trivial, simply choosing $g=sign(f)\cdot \frac{|f|^{p-1}}{\|f\|_p^{p-1}}$ gives that
\begin{align} \int_0^1 fg & = \frac{1}{\|f\|_p^{p-1}}\int_0^1 sign(f) \cdot f \cdot |f^{p-1}| = \frac{1}{\|f\|_p^{p-1}}\int_0^1 |f|^{p}= \\\\ & = \frac{\|f\|_p^p}{\|f\|^{p-1}} = \|f\|_p \end{align}
One can also show that $g\in L^q$ and $\|g\|_q=1$ by showing that
\begin{align} \int_0^1 |g|^q & = \frac{1}{\|f\|_p^{q(p-1)}}\int_0^1 |f^{p-1}|^q = \frac{1}{\|f\|_p^p}\int_0^1 |f|^{p}=1 \end{align}
I am stuck on showing the uniqueness of $g$. Any hints are appreciated.