PlanetMath: inner product

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inner product

(Definition)

An inner product on a vector space over a field (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 $\v _1, \v _2, \v , {{\bf w}}\in V$ , the following properties hold:

$(k_1 \v _1 + k_2 \v _2, {{\bf w}}) = k_1 (\v _1, {{\bf w}}) + k_2 (\v _2, {{\bf w}})$ (linearity ¹)
$(\v , {{\bf w}}) = \overline{({{\bf w}}, \v )}$ , where $\overline{\ \ \ \ }$ denotes complex conjugation (conjugate symmetry)
$(\v , \v ) \geq 0$ , and $(\v , \v ) = 0$ if and only if $\v = {{\bf0}}$ (positive definite)

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

The standard example of an inner product is the dot product on :

$\displaystyle ((x_1,\dots,x_n), (y_1,\dots,y_n)) := \sum_{i=1}^n x_i \overline{y_i}$

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

Footnotes

... (linearity ¹: A small minority of authors impose linearity on the second coordinate instead of the first coordinate.

"inner product" is owned by djao.

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Other names:

Hermitian inner product

Attachments:: semi-inner product (Definition) by asteroid

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Cross-references: norm, normed vector space, inner product space, dot product, even, inequality, positive definite, symmetry, conjugate, complex conjugation, coordinate, properties, function, complex numbers, real numbers, field, vector space
There are 103 references to this entry.

This is version 11 of inner product, born on 2002-01-24, modified 2006-10-22.
Object id is 1601, canonical name is InnerProduct.
Accessed 46127 times total.

Classification:

AMS MSC:	15A63 (Linear and multilinear algebra; matrix theory :: Quadratic and bilinear forms, inner products)
	11E39 (Number theory :: Forms and linear algebraic groups :: Bilinear and Hermitian forms)

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