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 $\u2102$ of complex numbers^{}) is a function $(,):V\times V\u27f6K$ such that, for all ${k}_{1},{k}_{2}\in K$ and ${\mathbf{v}}_{1},{\mathbf{v}}_{2},\mathbf{v},\mathbf{w}\in V$, the following properties hold:

1.
$({k}_{1}{\mathbf{v}}_{1}+{k}_{2}{\mathbf{v}}_{2},\mathbf{w})={k}_{1}({\mathbf{v}}_{1},\mathbf{w})+{k}_{2}({\mathbf{v}}_{2},\mathbf{w})$ (linearity^{1}^{1}A small minority of authors impose linearity on the second coordinate^{} instead of the first coordinate.)

2.
$(\mathbf{v},\mathbf{w})=\overline{(\mathbf{w},\mathbf{v})}$, where $\overline{}$ denotes complex conjugation (conjugate^{} symmetry^{})

3.
$(\mathbf{v},\mathbf{v})\ge 0$, and $(\mathbf{v},\mathbf{v})=0$ if and only if $\mathbf{v}=\mathrm{\U0001d7ce}$ (positive definite^{})
(Note: Rule 2 guarantees that $(\mathbf{v},\mathbf{v})\in \mathbb{R}$, so the inequality $(\mathbf{v},\mathbf{v})\ge 0$ in rule 3 makes sense even when $K=\u2102$.)
The standard example of an inner product is the dot product^{} on ${K}^{n}$:
$$(({x}_{1},\mathrm{\dots},{x}_{n}),({y}_{1},\mathrm{\dots},{y}_{n})):=\sum _{i=1}^{n}{x}_{i}\overline{{y}_{i}}$$ 
Every inner product space^{} is a normed vector space^{}, with the norm being defined by $\mathbf{v}:=\sqrt{(\mathbf{v},\mathbf{v})}$.
Title  inner product 

Canonical name  InnerProduct 
Date of creation  20130322 12:13:39 
Last modified on  20130322 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 