PlanetMath: Banach space

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Banach space

(Definition)

A Banach space $(X,\lVert \,\cdot\, \rVert )$ is a normed vector space such that is complete under the metric induced by the norm $\lVert \,\cdot\, \rVert$ .

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

If is a Banach space and is any normed vector space, then the set of continuous linear maps $f\colon X\to Y$ forms a Banach space, with norm given by the operator norm. In particular, since $\mathbb{R}$ and $\mathbb{C}$ 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:

Finite-dimensional normed vector spaces.
spaces are by far the most common example of Banach spaces.
$\ell^p$ spaces are spaces for the counting measure on $\mathbb{N}$ .
Continuous functions on a compact set under the supremum norm.
Finite signed measures on a $\sigma$ -algebra.

"Banach space" is owned by bbukh. [ full author list (2) | owner history (1) ]

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See Also: vector norm, dual space

Also defines:

dual space

Attachments:: Banach spaces of infinite dimension do not have a countable Hamel basis (Result) by yark
quotients of Banach spaces by closed subspaces are Banach spaces under the quotient norm (Theorem) by asteroid
necessary and sufficient conditions for a normed vector space to be a Banach space (Theorem) by asteroid

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Cross-references: signed measures, supremum norm, compact set, counting measure, linear functionals, operator norm, norm, linear maps, continuous, finite-dimensional, PlanetMath, infinite-dimensional, metric induced by the norm, complete, normed vector space
There are 80 references to this entry.

This is version 7 of Banach space, born on 2002-01-24, modified 2005-01-09.
Object id is 1605, canonical name is BanachSpace.
Accessed 21693 times total.

Classification:

AMS MSC:	46B99 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Miscellaneous)
	54E50 (General topology :: Spaces with richer structures :: Complete metric spaces)

Pending Errata and Addenda

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