- #1

Bipolarity

- 776

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A unitary operator from one vector space to another is one whose inverse and Hermitian transpose are identical.

It can be proved that unitary operators are norm preserving and inner product preserving. Which raises the question, which norm/inner product does this refer to, since the unitarity of the map was not defined with respect to an inner product?

In particular, I ask this question in application to signal processing.

Consider the space of functions that are integrable and square-integrable from -∞ to ∞. This space is a vector space, and also an inner product space under the standard function inner product. The norm induced by this inner product is the ##L^{2}## norm. It can be shown to be preserved under the Fourier transform, a result known as Parseval's theorem.

But does the Fourier transform preserve the ##L^{1}## norm, which does not appear to be induced by any inner product that I know of?

Thanks!

BiP