consider an infinite dimensional Hilbert space, e.g. the vector space of complex-valued functions with the inner product:
$\langle \psi, \phi \rangle = \int_{-\infty}^\infty \psi^{*}\phi dx \tag{1}$
Since this is a inner product, we need it to be finite. We apparently can guarantee that by forcing $\psi, \phi$ to be square-integrable since for a square integrable functon we know
$\int_{-\infty}^\infty |\psi |^2 dx < \infty \tag{3}$
Now, I don't really see how (3) leads to (1) being finite.
By Holder's /Cauchy -Schwarz inequality $\int |\psi^{*} \phi |dx \leq \sqrt {\int |\phi|^{2}dx}\sqrt {\int |\psi|^{2}dx} <\infty$.