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

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

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

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

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.

Example: For $N=3$ we have, in general,

$$\left<x,y\right> = x_0 \overline{y_0} + x_1 \overline{y_1} + x_2 \overline{y_2}.$$

Let

$$\begin{eqnarray*} x &=& [0,j,1] \\ y &=& [1,j,j]. \end{eqnarray*}$$

Then

$$\left<x,y\right> = 0\cdot 1 + j \cdot (-j) + 1 \cdot (-j) = 0 + 1 + (-j) = 1-j.$$