If X and Y are Banach spaces over the same ground field K, the set of all continuous K-linear maps T: X → Y is denoted by B ( X, Y ).

Banach spaces are a different generalization of Hilbert spaces.

The chain rule is also valid for Fréchet derivatives in Banach spaces.

* If is the norm ( usually noted as ) defined in the sequence space ℓ < sup >∞</ sup > of all bounded sequences ( which also matches the non-linear distance measured as the maximum of distances measured on projections into the base subspaces, without requiring the space to be isotropic or even just linear, but only continuous, such norm being definable on all Banach spaces ), and is lower triangular non-singular ( i. e., ) then

Condition numbers can be defined for any function ƒ mapping its data from some domain ( e. g. an m-tuple of real numbers x ) into some codomain an n-tuple of real numbers ƒ ( x ), where both the domain and codomain are Banach spaces.

In contrast, infinite-dimensional normed vector spaces may or may not be complete ; those that are complete are Banach spaces.

The fixed point theorem is often used to prove the inverse function theorem on complete metric spaces such as Banach spaces.

Later on, Stefan Banach amplified the concept, defining Banach spaces.

Many of the concepts in dynamical systems can be extended to infinite-dimensional manifolds — those that are locally Banach spaces — in which case the differential equations are partial differential equations.

The Euler – MacLaurin formula can be understood as a curious application of some ideas from Banach spaces and functional analysis.

Such spaces are called Banach spaces.

An important object of study in functional analysis are the continuous linear operators defined on Banach and Hilbert spaces.

General Banach spaces are more complicated than Hilbert spaces, and cannot be classified in such a simple manner as those.

In particular, Banach spaces lack a notion analogous to an orthonormal basis.

Examples of Banach spaces are-spaces for any real number.

The simplest way to define infinite dimensional Lie groups is to model them on Banach spaces, and in this case much of the basic theory is similar to that of finite dimensional Lie groups.

Of special interest are complete normed spaces called Banach spaces.

* James ` s Theorem For a Banach space the following two properties are equivalent:

Every Hilbert space X is a Banach space because, by definition, a Hilbert space is complete with respect to the norm associated with its inner product, where a norm and an inner product are said to be associated if for all x ∈ X.

The algebra multiplication and the Banach space norm are required to be related by the following inequality:

* Natural Banach function algebra: A uniform algebra whose all characters are evaluations at points of X.

Several elementary functions which are defined via power series may be defined in any unital Banach algebra ; examples include the exponential function and the trigonometric functions, and more generally any entire function.

* Permanently singular elements in Banach algebras are topological divisors of zero, i. e., considering extensions B of Banach algebras A some elements that are singular in the given algebra A have a multiplicative inverse element in a Banach algebra extension B. Topological divisors of zero in A are permanently singular in all Banach extension B of A.

( Gelfand – Naimark theorem ) Properties of the Banach space of continuous functions on a compact Hausdorff space are central to abstract analysis.

In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space.

It is named for Hans Hahn and Stefan Banach who proved this theorem independently in the late 1920s, although a special case was proved earlier ( in 1912 ) by Eduard Helly, and a general extension theorem from which the Hahn – Banach theorem can be derived was proved in 1923 by Marcel Riesz.

Some of the notable mathematical concepts named after Banach include Banach spaces, Banach algebras, the Banach – Tarski paradox, the Hahn – Banach theorem, the Banach – Steinhaus theorem, the Banach-Mazur game, the Banach – Alaoglu theorem and the Banach fixed-point theorem.

The theorem is named after Stefan Banach ( 1892 – 1945 ), and was first stated by him in 1922.

In January 2005 a New Jersey man named David Banach was arrested for pointing a green laser pointer at a small jet flying overhead.

In 1920, Banach was given an assistantship in Jagiellonian University after Poland regained independence.

While the school specialized in the humanities, Banach and his best friend Witold Wiłkosz ( also a future mathematician ) spent most of their time working on mathematics problems during breaks and after school.

In fact, soon after the encounter Steinhaus invited Banach to solve some problems he had been working on but which had proven difficult.

After the Red Army recaptured Lviv in the Lvov – Sandomierz Offensive of 1944, Banach returned to the University and helped re-establish it after the war years.

In this work Banach called such spaces " class E-spaces ", but in his 1932 book, Théorie des opérations linéaires, he changed terminology and referred to them as " spaces of type B ", which most likely contributed to the subsequent eponymous naming of these spaces after him.

If y is such that is a continuous linear functional on the domain of T, then, after extending it to the whole space via the Hahn – Banach theorem, we can find a z such that

Named after René Gâteaux, a French mathematician who died young in World War I, it is defined for functions between locally convex topological vector spaces such as Banach spaces.

Hadamard also founded the modern school of linear functional analysis further developed by Riesz and the group of Polish mathematicians around Stefan Banach.

Stefan Banach (; March 30, 1892 – August 31, 1945 ) was a Polish mathematician.

However, because the Soviets were removing Poles from annexed formerly Polish territories, Banach prepared to return to Krakòw.

In his early years Banach was tutored by Juliusz Mien, a French intellectual and friend of the Płowa family, who had emigrated to Poland and supported himself with photography and translations of Polish literature into French.

Steinhaus, Banach and Nikodym, along with several other Kraków mathematicians ( Władysław Ślebodziński, Leon Chwistek, Jan Kroć, and Włodzimierz Stożek ) also established a mathematical society, which eventually became the Polish Mathematical Society.

However, because the Soviets were removing Poles from annexed formerly Polish territories, Banach began preparing to leave the city and settle in Kraków, Poland, where he had been promised a chair at the Jagiellonian University.

* August 31 – Stefan Banach, Polish mathematician ( b. 1892 )

* March 30 – Stefan Banach, Polish mathematician ( d. 1945 )

Common examples of Polish spaces are the real line, any separable Banach space, the Cantor space, and Baire space.

* March 30-Stefan Banach ( died 1945 ), Polish mathematician.

* August 31-Stefan Banach ( born 1892 ), Polish mathematician.

* Czeslaw Olech, Director of International Mathematical Banach Centre, Member of the Polish Academy of Sciences, Professor, Warsaw University ( 1989 )

It was devised by a group of Polish mathematicians, Hugo Steinhaus, Bronisław Knaster and Stefan Banach, who used to meet in the Scottish Café in Lvov ( then in Poland ).

His work earned him an honored place in mathematics alongside such Polish mathematicians as Wojciech Brudzewski, Jan Brożek ( Broscius ), Nicolas Copernicus, Samuel Dickstein, Stefan Banach, Stefan Bergman, Marian Rejewski, Wacław Sierpiński, Stanisław Zaremba and Witold Hurewicz.

Hey is a Polish rock band founded in Szczecin in 1991 by guitarist Piotr Banach and lead singer Kasia Nosowska.

Outstanding Polish mathematicians formed the Lwów School of Mathematics ( with Stefan Banach, Hugo Steinhaus, Stanisław Ulam ) and Warsaw School of Mathematics ( with Alfred Tarski, Kazimierz Kuratowski, Wacław Sierpiński ).