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 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 continuous linear maps $f:X\to Y$ forms a Banach space, with norm given by the operator norm. In particular, since $ℝ$ and $ℂ$ are complete, the continuous linear functionals on a normed vector space $ℬ$ form a Banach space, known as the dual space of $ℬ$.

Examples:

• Finite-dimensional normed vector spaces (http://planetmath.org/EveryFiniteDimensionalNormedVectorSpaceIsABanachSpace).

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

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

• Continuous functions on a compact set under the supremum norm.

• Finite (http://planetmath.org/FiniteMeasureSpace) signed measures on a $\sigma$-algebra (http://planetmath.org/SigmaAlgebra).

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