Is this function well defined for any $g\in L^q(\Omega)$?

Question

If $p,q\in(1,\infty)$ such that $\frac 1q+\frac1p=1$, given $g\in L^q(\Omega)$ we difine: $$\Phi(g):L^p(\Omega)\to\Bbb R \\ \Phi(g)(f):= \int_{\Omega}fg$$ I know this is a basic question, but how do you show $\Phi$ is a well difined function for any $g\in L^q(\Omega)$? I mean, I'm getting confused, I thought that to show that a function is well defined you worked with the elements of the domain, in this case $L^p$.

Answer 1

By Holder's inequality, $$\|fg\|_{L^1} \le \|f\|_{L^p} \|g\|_{L^q}$$ Thus given $f \in L^p$ and $g \in L^q$, $fg \in L^1$, so $\int_{\Omega} fg$ makes sense as a real number.

Answer 2

I think it means for a fixed g $\in$ $L^q$, any f $\in L^p$, $\Phi(g)(f)= \int_{\Omega}fg$ is finite, thus well-defined. Using Holder's inequality, you can see.