Linear and multilinear algebra; matrix theory Definition

Inner product

Definition

Let $\mathbb{K}$ be $\R$ or $\C$ , and let $V$ be a vector space over $\mathbb{K}$ . An inner product on $V$ is a map $\langle \cdot, \cdot \rangle \colon V \times V \to \mathbb{K}$ such that, for all $x, y, z \in V$ and $a, b \in \mathbb{K}$ :

$\langle ax + by, z \rangle = a \langle x, z \rangle + b \langle y, z \rangle$ .
$\langle y, x \rangle = \overline{\langle x, y \rangle}$ .
$\langle x, x \rangle > 0$ if $x \ne 0$ .

Remarks

Over $\R$ , the conjugation does nothing, so an inner product is a symmetric positive definite bilinear form. Over $\C$ , conditions 1 and 2 give $\langle x, by \rangle = \bar b \langle x, y \rangle$ , so the map is conjugate linear in the second argument. Many physicists, and Mathlib, use the opposite convention: linear in the second argument. The dot product on $\R^n$ and $\langle x, y \rangle = \sum_i x_i \overline{y_i}$ on $\C^n$ are inner products.

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Vector space definition
Inner product definition

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Tag	`0VGSYBWHV2`
URI	`m2://0VGSYBWHV2`
Kind	definition
Writing	W1 stated
MSC	15A63, 46C05
Defines	Inner product
Uses	Vector space

Inner product

Definition

Remarks

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