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Etymologie

Was (wer) ist Banach space - definition

COMPLETE NORMED VECTOR SPACE

Complete normed vector space; Linear Algebra/Banach Spaces; Banach norm; Banach spaces; Banach function space; Complete normed space; Banach Space; Banach Spaces; Dual banach space; Complete norm; Isomorphic normed spaces

Banach space

<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. [Robert E. Megginson, "An Introduction to Banach Space Theory", Graduate Texts in Mathematics, 183, Springer Verlag, September 1998]. (2000-03-10)

Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

https://en.wikipedia.org/wiki/Banach_space

Multipliers and centralizers (Banach spaces)

Multiplier Banach space; Centralizer Banach space

In mathematics, multipliers and centralizers are algebraic objects in the study of Banach spaces. They are used, for example, in generalizations of the Banach–Stone theorem.

https://en.wikipedia.org/wiki/Multipliers_and_centralizers_(Banach_spaces)

Wikipedia

Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced [ˈbanax]) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space.

Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly.Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space." Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces.

https://en.wikipedia.org/wiki/Banach space