We have seen that the Fourier transform is a bijection on Schwartz space. In fact, it is a homeomorphism if we give its natural topology: the sub-basic sets are

Under this topology, is a completely metrizable LCS (locally convex space).

This LCS has many continuous linear functionals, they are called the **tempered distributions** . Note that any , , or can be identified with a distribution via

Other examples:

- is called the
**Dirac delta function**. - where is called the
**Cauchy principal value**.

Any continuous linear transformation on then defines a continuous linear map (the transpose) via , where .

ExampleAs is continuous on , its transpose “-” is continuous on .

ExampleSimilarly, the Fourier transform extends to a homeomorphism of because it is its own transpose (cf. below).

Lastly, note that

- is dense in .
- “Distributionally” means in the sense of distributions, more sociologically, it just means integrate against and follow your nose.

Examplein distributionally, i.e.

Lemma (Transpose of the Fourier Transform)For ,

i.e., the Fourier transform is its own transpose.

*Proof:* Fubini’s theorem gives

CorollaryIf then

Theorem (Plancherel)The Fourier transform extends continuous from to a unitary map on .

*Proof:* Suppose is a sequence in converges to in the -norm. By preceding Lemma,

which tends to as tends to . Namely, is a Cauchy sequence in , which is complete, and so converges. Let us call the limit . By intertwining sequences, for example, we see that the limit does not depend on the particular sequence .

Note that is an isometry because

Now, recall that unitary means an **onto** isometry. As the Fourier transform is an isometry, its range is closed and thus we just need to check that the range is dense. Well, is in the range, which is dense.

Lemma (Riemann–Lebesgue)If , then

- .
- is uniformly continuous.
- as .

*Proof:* All the assertions are true for . By the (obvious) inequality , these assertions are stable under taking limits.

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