Let $1\leq p<\infty$, $1/p+1/q=1$. For a function $f$ with $||f||_q=\infty$, can we write $$ ||f||_q=\sup_{g\in L^p(\Omega),||g||_p\neq 0}\frac{\int_\Omega |fg|}{||g||_p}? $$
or $$ ||f||_q=\sup_{||g||_p\leq 1}{\int_\Omega |fg|}? $$
In the case $||f||_q<\infty$, since $L^q(\Omega)$ is the dual space of $L^p(\Omega)$, with the $L^q$ norm equal to the norm of the bounded linear functional on $L^p$, we have
$$ ||f||_q=\sup_{||g||_p=1}{\int_\Omega |fg|}=\sup_{||g||_p\leq 1}{\int_\Omega |fg|}=\sup_{g\in L^p(\Omega),||g||_p\neq 0}\frac{\int_\Omega |fg|}{||g||_p}. $$
Thanks.
This is only true if you assume that $f$ vanishes outside a $\sigma$-finite set. For a counterexample, consider the measure space $X = \{1, 2\}$ with the measure $\mu$ defined on the power set of $X$ by $\mu(\emptyset) = 0$ and $\mu(\{1\}) = 1$ and $\mu(\{2\}) = \mu(X) = \infty$.
Then for any $g \in L^p (\Omega)$, we see that $g(2) = 0$. If we thus take $f$ defined by $f(2) = 1$ and $f(1) = 0$, then the right-hand side of your equality vanishes, but $\Vert f \Vert_p = \infty$.
So let us assume that $f \equiv 0$ on $\left(\bigcup_n M_n\right)^c$, where the $M_n$ have finite measure. By taking $M_n ' := \bigcup_{j=1}^n M_n$, we can assume w.l.o.g. that $M_n \subset M_{n+1}$ for all $n$.
Let us assume that the right-hand side is finite, i.e.
$$ C := \sup_{\Vert g \Vert_p \neq 0} \frac{\int_\Omega |f \cdot g|}{\Vert g \Vert_p} < \infty. $$
We will show that in this case also $\Vert f \Vert_q$ is finite. By contraposition, this proves your claim.
Now set $$K_n := \{x \mid |f(x)| \leq n\} \cap M_n.$$
Note that $K_n \subset K_{n+1}$ for all $n$ and $\bigcup_n K_n = \bigcup_n M_n$, so that $f$ vanishes on the complement of $\bigcup_n K_n$.
If we set $f_n := f \cdot \chi_{K_n}$, where $\chi_M$ denotes the indicator function of the set $M$, we get $f_n \rightarrow f$ pointwise as $n \rightarrow \infty$. It thus suffices (by Fatou's Lemma) to show that $\Vert f_n \Vert_q \leq C$ for all $n$.
To prove this, note that $f_n$ is bounded (by $n$) and vanishes outside a set of finite measure, so that $f_n \in L^q$. We can thus apply your characterization to conclude
$$ \Vert f_n \Vert_q = \sup_{\Vert g \Vert_p \neq 0} \frac{\int_\Omega |f_n \cdot g|}{\Vert g \Vert_p} = \sup_{\Vert g \Vert_p \neq 0} \frac{\int_\Omega |f \cdot (g \cdot \chi_{K_n})|}{\Vert g \Vert_p} \leq \sup_{\Vert g \Vert_p \neq 0} \frac{C \cdot \Vert g \cdot \chi_{K_n}\Vert_p}{\Vert g \Vert_p} \leq C. $$
Here, I used $\Vert g \cdot \chi_{K_n} \Vert_p \leq \Vert g \Vert_p$ in the last step and the definition of $C$ in the step before that.