Banach space
A Banach space^{} $(X,\parallel \cdot \parallel )$ is a normed vector space^{} such that $X$ is complete^{} under the metric induced by the norm $\parallel \cdot \parallel $.
Some authors use the term Banach space only in the case where $X$ is infinitedimensional, although on Planetmath finitedimensional 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 continuous^{} linear maps $f:X\to Y$ forms a Banach space, with norm given by the operator norm^{}. In particular, since $\mathbb{R}$ and $\u2102$ are complete, the continuous linear functionals^{} on a normed vector space $\mathcal{B}$ form a Banach space, known as the dual space^{} of $\mathcal{B}$.
Examples:

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Finitedimensional normed vector spaces (http://planetmath.org/EveryFiniteDimensionalNormedVectorSpaceIsABanachSpace).

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${L}^{p}$ spaces (http://planetmath.org/LpSpace) are by far the most common example of Banach spaces.

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${\mathrm{\ell}}^{p}$ spaces (http://planetmath.org/Lp) are ${L}^{p}$ spaces for the counting measure on $\mathbb{N}$.

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Continuous functions on a compact set under the supremum norm.

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Finite (http://planetmath.org/FiniteMeasureSpace) signed measures on a $\sigma $algebra (http://planetmath.org/SigmaAlgebra).
Title  Banach space 

Canonical name  BanachSpace 
Date of creation  20130322 12:13:48 
Last modified on  20130322 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 