For $f, g \in \mathcal{S}$: the inner product is usually given by $$ \langle f,g \rangle = \int f(x) g(x) dx $$
whereas for $f, g \in L^2$, the inner product is usually given by $$ \langle f,g \rangle = \int f(x) \overline{g(x)} dx $$
Given that the Schwartz space can be regarded as a subspace of $L^2$, how should interpret the difference between their inner products