Folland states Plancherel's theorem as follows: If $f \in L^1 \cap L^2$ then $\widehat{f} \in L^2$ and $\mathcal{F} | (L^1 \cap L^2)$ extends uniquely to a unitary isomorphism on $L^2$ where $\mathcal{F}$ stands for Schwartz functions.
I know we obtain Parseval's identity from this there, i.e $$ ||f||_2 = || \widehat{f}||_2$$ However, I am not entirely sure what is the significance of a unitary isomorphism.
For example, if $f \in \mathcal{H} \subset L^2$, where $\mathcal{H}$ is some closed subspace of the Hilbert space $L^2$ If I find a basis for the fourier transforms $\widehat{f} \in L^2$, how does this related to a basis for $\mathcal{H}$?
Sorry if the above is confusing. I am just not sure as to how to interpret the significance of Plancherel's theorem.
Unitary means it preserves the inner product so "unitary isomorphism" is a fancy way of saying that the Fourier transform is an isometry of $L^2$.
Morally, Plancherel's theorem says that you can think of the Fourier transform as like a 'rotation' of $L^2$. It preserves distances and angles, as measured by the $L^2$ inner product.