PlanetMath: inner product space

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inner product space

(Definition)

An inner product space (or pre-Hilbert space) is a vector space (over $\mathbb{R}$ or $\mathbb{C}$ ) with an inner product ${\langle \cdot,\cdot\rangle}$ .

For example, $\mathbb{R}^n$ with the familiar dot product forms an inner product space.

Every inner product space is also a normed vector space, with the norm defined by $\Vert x \Vert := \sqrt{{\langle x,\,x\rangle}}$ . This norm satisfies the parallelogram law.

If the metric $\Vert{x-y}\Vert$ induced by the norm is complete, then the inner product space is called a Hilbert space.

The Cauchy-Schwarz inequality

$\displaystyle \vert{\langle x,\,y\rangle}\vert \le \Vert x\Vert \cdot\Vert y\Vert$

(1)

holds in any inner product space.

According to (1), one can define the angle between two non-zero vectors and :

$\displaystyle \cos(x,\,y) := \frac{{\langle x,\,y\rangle}}{\Vert{x}\Vert\cdot\Vert{y}\Vert}.$

(2)

This provides that the scalars are the real numbers. In any case, the perpendiculatity of the vectors may be defined with the condition

$\displaystyle {\langle x,\,y\rangle} =0.$

"inner product space" is owned by CWoo. [ full author list (4) | owner history (2) ]

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Other names:

pre-Hilbert space

Also defines:

angle between two vectors, perpendicularity

Attachments:: Euclidean vector space (Definition) by rmilson

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Cross-references: vectors, real numbers, scalars, non-zero vectors, angle, inequality, Hilbert space, metric, parallelogram law, normed vector space, dot product, inner product, vector space
There are 35 references to this entry.

This is version 19 of inner product space, born on 2002-01-24, modified 2007-10-06.
Object id is 1613, canonical name is InnerProductSpace.
Accessed 33497 times total.

Classification:

AMS MSC:

46C99 (Functional analysis :: Inner product spaces and their generalizations, Hilbert spaces :: Miscellaneous)

Pending Errata and Addenda

None.[ View all 6 ]

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