banach space

Banach space

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Definitions of banach space words

noun banach space a vector space on which a norm is defined that is complete. 1
noun Definition of banach space in Technology (mathematics) A complete normed vector space. Metric is induced by the norm: d(x,y) = ||x-y||. Completeness means that every Cauchy sequence converges to an element of the space. All finite-dimensional real and complex normed vector spaces are complete and thus are Banach spaces. Using absolute value for the norm, the real numbers are a Banach space whereas the rationals are not. This is because there are sequences of rationals that converges to irrationals. Several theorems hold only in Banach spaces, e.g. the Banach inverse mapping theorem. All finite-dimensional real and complex vector spaces are Banach spaces. Hilbert spaces, spaces of integrable functions, and spaces of absolutely convergent series are examples of infinite-dimensional Banach spaces. Applications include wavelets, signal processing, and radar. 1
noun banach space (analysis) A normed vector space which is complete, in the sense that Cauchy sequences have limits. 0

before 1945

One of the 6% newest English words

1945-50; after Stefan Banach (1892-1945), Polish mathematician

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This term is known only to a narrow circle of people with rare knowledge. Only 27% of English native speakers know the meaning of this word.

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