Banach space

A Banach spaceMathworldPlanetmath (X,) is a normed vector spacePlanetmathPlanetmath such that X is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath under the metric induced by the norm .

Some authors use the term Banach space only in the case where X is infinite-dimensional, although on Planetmath finite-dimensional spaces are also considered to be Banach spaces.

If Y is a Banach space and X is any normed vector space, then the set of continuousMathworldPlanetmathPlanetmath linear maps f:XY forms a Banach space, with norm given by the operator normMathworldPlanetmath. In particular, since and are complete, the continuous linear functionalsMathworldPlanetmathPlanetmath on a normed vector space form a Banach space, known as the dual spaceMathworldPlanetmathPlanetmath of .


  • Finite-dimensional normed vector spaces (

  • Lp spaces ( are by far the most common example of Banach spaces.

  • p spaces ( are Lp spaces for the counting measure on .

  • Finite ( signed measures on a σ-algebra (

Title Banach space
Canonical name BanachSpace
Date of creation 2013-03-22 12:13:48
Last modified on 2013-03-22 12:13:48
Owner bbukh (348)
Last modified by bbukh (348)
Numerical id 11
Author bbukh (348)
Entry type Definition
Classification msc 46B99
Classification msc 54E50
Related topic VectorNorm
Related topic DualSpace
Defines dual space