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...

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...

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

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

"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...

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...

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

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

one shows that each Vitali set has Banach measure 0. This does not create any contradictions since Banach measures are not countably additive, but only...

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...

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

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

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...

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...

σ-algebra which is often used when studying product measures or probability measures of random variables on Banach spaces. For a product space, the cylinder σ-algebra...

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....

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

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 the mathematical field of functional analysis, Banach spaces are among the most important objects of study. In other areas of mathematical analysis...

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...

the push-forward and the standard Gaussian measure on the real line: a Borel measure γ on a separable Banach space X is called Gaussian if the push-forward...

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...

The Banach–Tarski Paradox is a book in mathematics on the Banach–Tarski paradox, the fact that a unit ball can be partitioned into a finite number of subsets...

analysis, a Banach limit is a continuous linear functional ϕ : ℓ ∞ → C {\displaystyle \phi :\ell ^{\infty }\to \mathbb {C} } defined on the Banach space ℓ...

Real analysis. Addison-Wesley. Contains a proof for vector measures assuming values in a Banach space. Royden, H. L.; Fitzpatrick, P. M. (2010). Real Analysis...

objects overlap. It was proposed by Hugo Steinhaus and proved by Stefan Banach (explicitly in dimension 3, without taking the trouble to state the theorem...

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

C ( [ 0 , 1 ] ) {\displaystyle C([0,1])} . The Banach–Mazur theorem asserts that any separable Banach space is isometrically isomorphic to a closed linear...