The Inner Product

The inner product (or ``dot 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 $\{{\bf C}^N,{\bf C}\}$ for this book (complex length $N$ vectors, and complex scalars).

The inner product between (complex) $N$-vectors $x$ and $y$ is defined by^5.6

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

The complex conjugation of the second vector is done in order that a norm will be induced by the inner product:^5.7

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

As a result, the inner product is conjugate symmetric:

$$\left<y,x\right> = \overline{\left<x,y\right>}$$

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