inner product space
An inner product space (or pre-Hilbert space) is a vector space
(over or )with an inner product
.
For example, with the familiar dot productforms an inner product space.
Every inner product space is also a normed vector space,with the norm defined by .This norm satisfies the parallelogram law
.
If the metric induced by the norm is complete (http://planetmath.org/Complete),then the inner product space is called a Hilbert space.
The Cauchy–Schwarz inequality
(1) |
holds in any inner product space.
According to (1), one can define the angle between two non-zero vectors and :
(2) |
This provides that the scalars are the real numbers. In any case, the perpendiculatity of the vectors may be defined with the condition
Title | inner product space |
Canonical name | InnerProductSpace |
Date of creation | 2013-03-22 12:14:05 |
Last modified on | 2013-03-22 12:14:05 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 23 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 46C99 |
Synonym | pre-Hilbert space |
Related topic | InnerProduct |
Related topic | OrthonormalBasis |
Related topic | HilbertSpace |
Related topic | EuclideanVectorSpace2 |
Related topic | AngleBetweenTwoLines |
Related topic | FluxOfVectorField |
Related topic | CauchySchwarzInequality |
Defines | angle between two vectors |
Defines | perpendicularity |