inner product

An inner product on a vector space $V$ over a field $K$ (which must be either the field $\mathbb{R}$ of real numbers or the field $\mathbb{C}$ of complex numbers) is a function $(\ ,\ ):V\times V\longrightarrow K$ such that, for all $k_{1},k_{2}\in K$ and ${{\bf v}}_{1},{{\bf v}}_{2},{{\bf v}},{{\bf w}}\in V$ , the following properties hold:

1.

$(k_{1}{{\bf v}}_{1}+k_{2}{{\bf v}}_{2},{{\bf w}})=k_{1}({{\bf v}}_{1},{{\bf w}% })+k_{2}({{\bf v}}_{2},{{\bf w}})$ (linearity¹¹A small minority of authors impose linearity on the second coordinate instead of the first coordinate.)
2.

$({{\bf v}},{{\bf w}})=\overline{({{\bf w}},{{\bf v}})}$ , where $\overline{\ \ \ \ }$ denotes complex conjugation (conjugate symmetry)
3.

$({{\bf v}},{{\bf v}})\geq 0$ , and $({{\bf v}},{{\bf v}})=0$ if and only if ${{\bf v}}={{\bf 0}}$ (positive definite)

(Note: Rule 2 guarantees that $({{\bf v}},{{\bf v}})\in\mathbb{R}$ , so the inequality $({{\bf v}},{{\bf v}})\geq 0$ in rule 3 makes sense even when $K=\mathbb{C}$ .)

The standard example of an inner product is the dot product on $K^{n}$ :

$((x_{1},\dots,x_{n}),(y_{1},\dots,y_{n})):=\sum_{i=1}^{n}x_{i}\overline{y_{i}}$

Every inner product space is a normed vector space, with the norm being defined by $||{{\bf v}}||:=\sqrt{({{\bf v}},{{\bf v}})}$ .

Title	inner product
Canonical name	InnerProduct
Date of creation	2013-03-22 12:13:39
Last modified on	2013-03-22 12:13:39
Owner	djao (24)
Last modified by	djao (24)
Numerical id	15
Author	djao (24)
Entry type	Definition
Classification	msc 11E39
Classification	msc 15A63
Synonym	Hermitian inner product
Related topic	InnerProductSpace
Related topic	HermitianForm
Related topic	EuclideanVectorSpace