inner product space

In mathematics, a vector space or function space in which an operation for combining two vectors or functions (whose result is called an inner product) is defined and has certain properties. Such spaces, an essential tool of functional analysis and vector theory, allow analysis of classes of functions rather than individual functions. In mathematical analysis, an inner product space of particular importance is a Hilbert space, a generalization of ordinary space to an infinite number of dimensions. A point in a Hilbert space can be represented as an infinite sequence of coordinates or as a vector with infinitely many components. The inner product of two such vectors is the sum of the products of corresponding coordinates. When such an inner product is zero, the vectors are said to be orthogonal (see orthogonality). Hilbert spaces are an essential tool of mathematical physics. See also David Hilbert.

inner product space [¦in·ər ′präd·əkt ‚spās]

(mathematics)

A vector space that has an inner product defined on it. Also known as generalized Euclidean space; Hermitian space; pre-Hilbert space.

Mentioned in	?		References in periodicals archive	?		Encyclopedia browser	?		Full browser	?
No references found	d]) as a real inner product space with inner product (S, T) = ([phi](S), [phi](T)) = Tr([phi](S)[phi](T)'). Wishartness of quadratic forms: a characterization via Jordan algebra ... by Masaro, Joe; Wong, Chi Song / Tamsui Oxford Journal of Mathematical Sciences The authors (both professors of mathematics at the Rose- Hulman Institute of Technology) develop the mathematical framework of vector and inner product spaces upon which signal and image processing depends; develop traditional Fourier-based transform techniques, primarily in the discrete case, but also somewhat in the continuous setting; and provide entry-level material on filtering, convolution, filter banks, and wavelets. Discrete fourier analysis and wavelets; applications to signal and ... by SciTech Book News 2] can be turned into an inner product space by defining [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and extending by linearity. The tensor product of frames by Bourouihiya, Abdelkrim / Sampling Theory in Signal and Image Processing	inner bottom inner bremsstrahlung inner casing inner cell mass inner child inner class inner core inner court Inner Ear Inner Form inner function inner harbor inner hearth Inner Hebrides inner join inner keel inner Lagrangian point inner mantle inner marker inner measure Inner Mongolia Inner Mongolian Autonomous Region inner planet inner potential inner product inner product space inner quantum number inner sanctum Inner Speech inner strake Inner Temple inner tube inner-tube extension Innervation Innes, James Innes, Michael Inness, George innings Innis, Harold Adams Innis, Roy innkeeper's daughter Innocence Innocent I, Saint Innocent II Innocent III Innocent IV Innocent V Innocent VI Innocent VIII Innocent XI Innocent XII	Inner Mongolian Inner Mongolian Autonomous Region Inner Mongolian Autonomous Region Inner Mongolian Autonomous Region Inner Mongolian People's Party Inner Moray Firth Inner Moray Firth inner nature Inner normal Inner North Brisbane Mental Health Service Inner nuclear layer Inner nuclear layer Inner nuclear layer Inner nuclear layer Inner Oort cloud Inner Oort cloud Inner Parish Security Corporation inner part Inner Peace Television Network Inner Physical Layer inner planet inner planet inner planets inner planets Inner plate Inner post inner potential inner product inner product Inner Product of Vectors inner product space Inner product spaces Inner products Inner psyche Inner psyche inner quantum number Inner Radiation Belt Inner Radius inner reality inner resource inner retinal nuclear layer inner retinal nuclear layer inner retinal nuclear layer Inner ring Inner Ring Road Maintenance Scheme Inner River Zone Inner salt Inner salt Inner salt Inner Sanctum Inner Sanctum Mysteries inner sanctums Inner self Inner Self-Helper inner sole Inner South Community Health Service Inner South East Partnership in Community and Health Inner space Inner space (disambiguation) Inner Space Interactive Sourcing

Mentioned in

References in periodicals archive

Encyclopedia browser

Full browser

No references found

d]) as a real inner product space with inner product (S, T) = ([phi](S), [phi](T)) = Tr([phi](S)[phi](T)').

Wishartness of quadratic forms: a characterization via Jordan algebra ... by Masaro, Joe; Wong, Chi Song / Tamsui Oxford Journal of Mathematical Sciences

The authors (both professors of mathematics at the Rose- Hulman Institute of Technology) develop the mathematical framework of vector and inner product spaces upon which signal and image processing depends; develop traditional Fourier-based transform techniques, primarily in the discrete case, but also somewhat in the continuous setting; and provide entry-level material on filtering, convolution, filter banks, and wavelets.

Discrete fourier analysis and wavelets; applications to signal and ... by SciTech Book News

2] can be turned into an inner product space by defining [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and extending by linearity.

The tensor product of frames by Bourouihiya, Abdelkrim / Sampling Theory in Signal and Image Processing

Inner Mongolian
Inner Mongolian Autonomous Region
Inner Mongolian Autonomous Region
Inner Mongolian Autonomous Region
Inner Mongolian People's Party
Inner Moray Firth
Inner Moray Firth
inner nature
Inner normal
Inner North Brisbane Mental Health Service
Inner nuclear layer
Inner nuclear layer
Inner nuclear layer
Inner nuclear layer
Inner Oort cloud
Inner Oort cloud
Inner Parish Security Corporation
inner part
Inner Peace Television Network
Inner Physical Layer
inner planet
inner planet
inner planets
inner planets
Inner plate
Inner post
inner potential
inner product
inner product
Inner Product of Vectors

inner product space

Inner product spaces
Inner products
Inner psyche
Inner psyche
inner quantum number
Inner Radiation Belt
Inner Radius
inner reality
inner resource
inner retinal nuclear layer
inner retinal nuclear layer
inner retinal nuclear layer
Inner ring
Inner Ring Road Maintenance Scheme
Inner River Zone
Inner salt
Inner salt
Inner salt
Inner Sanctum
Inner Sanctum Mysteries
inner sanctums
Inner self
Inner Self-Helper
inner sole
Inner South Community Health Service
Inner South East Partnership in Community and Health
Inner space
Inner space (disambiguation)
Inner Space Interactive Sourcing