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. 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. 2.

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

3. 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 InnerProduct 2013-03-22 12:13:39 2013-03-22 12:13:39 djao (24) djao (24) 15 djao (24) Definition msc 11E39 msc 15A63 Hermitian inner product InnerProductSpace HermitianForm EuclideanVectorSpace