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 by^5.9

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

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

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

As a result, the inner product is conjugate symmetric:

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

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.