In the definition of the Hilbert space, it says it consists of all the vectors that are norm-square integrable, or
$$ \int_a^b |f(x)|^2 dx < \infty. $$
However, I'm unsure if the $|f(x)|^2$ is computed by the inner product of the Hilbert space
$$ <f|g> = \int_a^b f(x)^* g(x) dx $$
or if it's simply squared.
I think the confusion stems from the norm also being written as $|f(x)|^2$ which is computed by taking the inner product with itself $<f|f>$.