Linearity of the Inner Product

Any function $f(x)$ of a vector $x\in{\bf C}^N$ (which we may call an operator on ${\bf C}^N$) is said to be linear if for all $x_1\in{\bf C}^N$ and $x_2\in{\bf C}^N$, and for all scalars $c_1$ and $c_2$ in ${\bf C}$, we have

$$f(c_1 x_1 + c_2 x_2) = c_1 f(x_1) + c_2 f(x_2).$$

A linear operator thus ``commutes with mixing.''

Linearity consists of two component properties,

• additivity: $f(x_1+x_2) = f(x_1) + f(x_2)$, and
• homogeneity: $f(c_1 x_1) = c_1 f(x_1)$.

A function of multiple vectors, e.g., $f(x,y,z)$ can be linear or not with respect to each of its arguments.

The inner product $\left<x,y\right>$ is linear in its first argument, i.e.,

$$\left<c_1 x_1 + c_2 x_2,y\right> = c_1 \left<x_1,y\right> + c_2 \left<x_2,y\right>.$$

This is easy to show from the definition:

$$\begin{eqnarray*} \left<c_1 x_1 + c_2 x_2,y\right> &\isdef & \sum_{n=0}^{N-1}\le... ...)} \\ &\isdef & c_1 \left<x_1,y\right> + c_2 \left<x_2,y\right> \end{eqnarray*}$$

The inner product is also additive in its second argument, i.e.,

$$\left<x,y_1 + y_2\right> = \left<x,y_1\right> + \left<x,y_2\right>,$$

but it is only conjugate homogeneous in its second argument, since

$$\left<x,c_1 y_1\right> = \overline{c_1} \left<x,y_1\right> \neq c_1 \left<x,y_1\right>.$$

The inner product is strictly linear in its second argument with respect to real scalars:

$$\left<x,r_1 y_1 + r_2 y_2\right> = r_1 \left<x,y_1\right> + r_2 \left<x,y_2\right>, \quad r_i\in{\bf R}$$

Since the inner product is linear in both of its arguments for real scalars, it is often called a bilinear operator in that context.