Inner product - Knowlpedia

An inner product on a vector space $V$ over $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$ is a map

\langle\cdot,\cdot\rangle:V\times V\to\mathbb{F}

such that for all $u,v,w\in V$ and $a,b\in\mathbb{F}$ :

\langle au+bw,\,v\rangle=a\langle u,v\rangle+b\langle w,v\rangle,\qquad \langle u,v\rangle=\overline{\langle v,u\rangle},\qquad \langle v,v\rangle\ge 0\ \text{and }\langle v,v\rangle=0\iff v=0.

Remarks

This convention is linear in the first argument and conjugate-linear in the second. In particular, $\langle v,v\rangle$ is real, so positivity is an inequality of real numbers.

An inner product is closely related to a bilinear form (or sesquilinear form in the complex case). It induces a norm via $\|v\|=\sqrt{\langle v,v\rangle}$ and defines orthogonality through the condition $\langle u,v\rangle=0$ .

Examples

On $\mathbb{R}^n$ , the standard dot product $\langle x,y\rangle=\sum_{i=1}^n x_i y_i$ is an inner product.
On $\mathbb{C}^n$ , the Hermitian product $\langle x,y\rangle=\sum_{i=1}^n x_i\overline{y_i}$ is an inner product.
If $W$ is a symmetric positive-definite matrix, then $\langle x,y\rangle=x^\mathsf{T}Wy$ defines an inner product on $\mathbb{R}^n$ .