Does it make sense to talk about $L^2$ inner product of two functions not necessarily in $L^2$? The $L^2$-inner product of two real functions $f$ and $g$ on a measure space $X$ with respect to the measure $\mu$ is given by
$$
\langle f,g\rangle_{L^2} := \int_X fg d\mu, $$
When $f$ and $g$ are both in $L^2(X)$, $|\langle f,g\rangle_{L^2}|\leqslant \|f\|_{L^2} \|g\|_{L^2} < \infty$.
I was wondering if it makes sense to talk about $\langle f,g\rangle_{L^2}$ when $f$ and/or $g$ may not be in  $L^2(X)$? What are cases more general than $f$ and $g$  both in $L^2(X)$, when talking about  $L^2(X)$ makes sense?
Thanks and regards!

Yes it does make sense. For example, if you take $f\in L^p(X)$ and $g\in L^{p'}(X)$ with $\frac{1}{p}+\frac{1}{p'}=1$, then $\langle f,g\rangle_{L^2} $ is well defined and $$\langle f,g\rangle_{L^2} <\|f\|_p\|g\|_{p'}$$
With this notation it is said the the inner product on $L^2$ induces the duality $p$ and $p'$.
