Banach space
A Banach space (X,∥⋅∥) is a normed vector space
such that X is complete
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 continuous linear maps f:X→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:
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•
Finite-dimensional normed vector spaces (http://planetmath.org/EveryFiniteDimensionalNormedVectorSpaceIsABanachSpace).
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Lp spaces (http://planetmath.org/LpSpace) are by far the most common example of Banach spaces.
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ℓp spaces (http://planetmath.org/Lp) are Lp spaces for the counting measure on 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 σ-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 |