This post’ll be pretty quick—the Plancherel theorem, a basic result on Fourier transforms, is a quick corollary of what I’ve already talked about.

We have shown that the Fourier transform is an isomorphism of onto itself; the inverse is given by the inverse Fourier transform. The next step is to extend this to an isometry of onto itself. Since is dense in , it will be sufficient to show that

for . We will do this by proving the identity

for . This is a simple computation:

In other words, the Fourier transform and its inverse are adjoints. If we take and use the inversion formula, it becomes clear that the Fourier transform preserves the -norm, whence follows

**Theorem 1 (Plancherel)** *The Fourier transform extends to an isometry of onto itself. *

Incidentally, for a function , it is **not** necessarily true that

because that integral need not exist. It is, however, true that the integral will exist almost everywhere in a “principal value” sense, which we do not need to bother with here.

In a sense, this is a continuous analog of the Parseval theorem, which states that the Fourier coefficient map from (for the space of double-sided, square-summable sequences) is an isometry.

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