This is a follow up question to this question:
Let $\|f(t)\|_p = (\int^b_a|f(t)|^pdt)^{1/p}$ for continuous $\Bbb{R}$-valued functions on $(a,b)$. Show that if $\frac{1}{p}+\frac{1}{q}=1$ then $$\tag{1}\|f(t)\|_p=\sup_{\|g\|_q=1}\int f(t)g(t)dt.$$ Conclude then that $\|\cdot\|_p$ is a norm.
I can see that $\|f(t)\|_p$ is the same as $\|f(t)\|_2$ where $p=2$, so I believe the proof to show this is a norm would be similar to how you show $\|f(t)\|_2$ is a norm, but $(1)$ has really got me confused. Any help would be greatly appreciated!