In the mathematical discipline of measure theory, a Banach measure is a certain way to assign a size (or area) to all subsets of the Euclidean plane, consistent...

The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists...

that bear Banach's name include Banach spaces, Banach algebras, Banach measures, the Banach–Tarski paradox, the Hahn–Banach theorem, the Banach–Steinhaus...

hence complex measures include finite signed measures but not, for example, the Lebesgue measure. Measures that take values in Banach spaces have been...

that each Vitali set has Banach measure 0 {\displaystyle 0} . This does not create any contradictions since Banach measures are not countably additive...

"area". (This Banach measure, however, is only finitely additive, so it is not a measure in the full sense, but it equals the Lebesgue measure on sets for...

functional analysis, a Banach space (/ˈbɑː.nʌx/, Polish pronunciation: [ˈba.nax]) is a complete normed vector space. Thus, a Banach space is a vector space...

In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A {\displaystyle A} over the real...

The concentration of measure phenomenon was put forth in the early 1970s by Vitali Milman in his works on the local theory of Banach spaces, extending an...

In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a...

Lebesgue measure is a measure defined on infinite-dimensional normed vector spaces, such as Banach spaces, which resembles the Lebesgue measure used in...

Examples of Banach spaces are L p {\displaystyle L^{p}} -spaces for any real number p ≥ 1 {\displaystyle p\geq 1} . Given also a measure μ {\displaystyle...

class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability...

weak convergence can be extended to Banach spaces. A sequence of points ( x n ) {\displaystyle (x_{n})} in a Banach space B is said to converge weakly...

In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral...

variation and has the Luzin N property. This statement is also known as the Banach-Zareckiǐ theorem. If f: I → R is absolutely continuous and g: R → R is globally...

measure space fulfills the conditions of being localizable and therefore semifinite). Pointwise multiplication gives them the structure of a Banach algebra...

canonical Gaussian cylinder set measure on H. H. Satô, Gaussian Measure on a Banach Space and Abstract Wiener Measure, 1969. Dudley, Richard M.; Feldman...

In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff...

measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by convergence of measures,...

cylinder set Gaussian measure on a separable Hilbert space and chooses a Banach space in such a way that the cylindrical measure becomes σ-additive on...

geometry) Banach fixed-point theorem Banach game Banach lattice Banach limit Banach manifold Banach measure Banach space Banach coordinate space Banach disks...

ISBN 978-3-540-64563-4. OCLC 246032063. Oxtoby, John C. (1980). "The Banach Category Theorem". Measure and Category (Second ed.). New York: Springer. pp. 62–65....

infinite dimensional Lebesgue measure. The notion of an abstract Wiener space allows one to construct a measure on a Banach space B that contains a Hilbert...

happens for example when μ {\displaystyle \mu } is a measure on a finite set). Likewise, the Banach space C ( [ 0 , 1 ] ) {\displaystyle C([0,1])} of continuous...

In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball...

decomposition into finitely many pieces must preserve the sum of the Banach measures of the pieces, and therefore cannot change the total area of a set...

finite signed measures becomes a Banach space. This space has even more structure, in that it can be shown to be a Dedekind complete Banach lattice and...

sets Σ {\displaystyle \Sigma } is the Banach space consisting of all bounded and finitely additive signed measures on Σ {\displaystyle \Sigma } . The norm...

finite-dimensional vector spaces. Every finite-dimensional subspace of a Banach space is complemented, but other subspaces may not. In general, classifying...