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The Inner Product

The inner product (or ``dot product'', or ``scalar product'') is an operation on two vectors which produces a scalar. Defining an inner product for a Banach space specializes it to a Hilbert space (or ``inner product space''). There are many examples of Hilbert spaces, but we will only need $ \{\mathbb{C}^N,\mathbb{C}\}$ for this book (complex length-$ N$ vectors, and complex scalars).

The inner product between (complex) $ N$ -vectors $ \underline{u}$ and $ \underline{v}$ is defined by5.9

$\displaystyle \zbox {\left<\underline{u},\underline{v}\right> \isdef \sum_{n=0}^{N-1}u(n)\overline{v(n)}.}

The complex conjugation of the second vector is done in order that a norm will be induced by the inner product:5.10

$\displaystyle \left<\underline{u},\underline{u}\right> = \sum_{n=0}^{N-1}u(n)\overline{u(n)}
= \sum_{n=0}^{N-1}\left\vert u(n)\right\vert^2 \isdef {\cal E}_u = \Vert u\Vert^2

As a result, the inner product is conjugate symmetric:

$\displaystyle \left<\underline{v},\underline{u}\right> = \overline{\left<\underline{u},\underline{v}\right>}

Note that the inner product takes $ \mathbb{C}^N\times\mathbb{C}^N$ to $ \mathbb{C}$ . That is, two length $ N$ complex vectors are mapped to a complex scalar.

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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8
Copyright © 2024-04-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University