The term-algebra was introduced by C. E. Rickart in 1946 to describe Banach *- algebras that satisfy the condition:

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.

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

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

Stefan Banach was born on 30 March 1892 at St. Lazarus General Hospital in Kraków, then part of Austro-Hungarian Empire.

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.

While generally Banach was a diligent student he did on occasion receive low grades ( he failed Greek during his first semester at the gymnasium ) and would later speak critically of the school's math teachers.

As Banach had to earn money to support his studies it was not until 1914 that he finally, at age 22, passed his high school graduation exams.

When World War I broke out, Banach was excused from military service due to his left-handedness and poor vision.

It was also through Steinhaus that Banach met his future wife, Łucja Braus.

After Poland regained independence, in 1920 Banach was given an assistantship at Kraków's Jagiellonian University.

The book was also the first in a long series of mathematics monographs edited by Banach and his circle.

Following the German takeover of Lwów in 1941 during Operation Barbarossa, all universities were closed and Banach, along with many colleagues and his son, was employed as lice feeder at Professor Rudolf Weigl's Typhus Research Institute.

However, because Wiener's work on the topic was limited, the established name became just Banach spaces.

Likewise, Banach's fixed point theorem, based on earlier methods developed by Charles Émile Picard, was included in his dissertation, and was later extended by his students ( for example in the Banach – Schauder theorem ) and other mathematicians ( in particular Bouwer and Poincaré and Birkhoff ).

: " Banach was my greatest scientific discovery.

Within two years, Józef Ulam and the rest of his family were victims of the Holocaust, Steinhaus was in hiding, Kuratowski was lecturing at the underground university in Warsaw, Stożek and his two sons had been killed in the massacre of Lwów professors, Banach was surviving Nazi occupation by feeding lice at Rudolf Weigl's typhus research institute, and the last problem had been recorded in the Scottish Book.

Kuratowski ’ s research in the field of measure theory, including research with Banach, Tarski, was continued by many students.

For example, one cannot define all the trigonometric functions in a Banach algebra without identity.

When the Banach algebra A is the algebra L ( X ) of bounded linear operators on a complex Banach space X ( e. g., the algebra of square matrices ), the notion of the spectrum in A coincides with the usual one in the operator theory.

This case and the previous one admit a simultaneous generalization to Banach manifolds.

A further generalization for a function from one Banach space to another is the Fréchet derivative.

In 1916, in Kraków's Planty gardens, Banach encountered Professor Hugo Steinhaus, one of the renowned mathematicians of the time.

The Hahn – Banach theorem, is one of fundamental theorems of functional analysis.

In mathematics, the uniform boundedness principle or Banach – Steinhaus theorem is one of the fundamental results in functional analysis.

Together with the Hahn – Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field.

For example, when the space of functions is a Banach space, the functional derivative becomes known as the Fréchet derivative, while one uses the Gâteaux derivative on more general locally convex spaces.

By the Krein – Milman theorem one can show without too much difficulty that for x an element of the Banach *- algebra A having an approximate identity:

One of Gelfand's original applications ( and one which historically motivated much of the study of Banach algebras ) was to give a much shorter and more conceptual proof of a celebrated lemma of Norbert Wiener ( see the citation below ), characterizing the elements of the group algebras L < sup > 1 </ sup >( R ) and whose translates span dense subspaces in the respective algebras.

This equivalence also serves to demonstrate the necessity of X and Y being Banach ; one can construct linear maps that have unbounded inverses in this setting, for example, by using either continuous functions with compact support or by using sequences with finitely many non-zero terms along with the supremum norm.

* K ( X, Y ) is a closed subspace of B ( X, Y ): Let T < sub > n </ sub >, n ∈ N, be a sequence of compact operators from one Banach space to the other, and suppose that T < sub > n </ sub > converges to T with respect to the operator norm.

The Banach – Tarski paradox shows that there is no way to define volume in three dimensions unless one of the following four concessions is made:

The Schauder fixed-point theorem states, in one version, that if C is a nonempty closed convex subset of a Banach space V and f is a continuous map from C to C whose image is compact, then f has a fixed point.

This proof follows the one described by Steinhaus and others ( 1938 ), attributed there to Stefan Banach, for the case.

Calderon's method enables one to construct a family of new Banach spaces,

Detail from one of the 353 memorial poles, by Tamkin Noorzad, Ebru Agim, Ashley Banach and Amie Lozenkovski of Keira High School, Wollongong

In the more general case of Banach spaces, one has

A first step in extending the theory of holomorphic functions beyond one complex dimension is considering so-called vector-valued holomorphic functions, which are still defined in the complex plane C, but take values in a Banach space.

By defining suitable norms and / or inner products, one can exhibit sets of harmonic functions which form Hilbert or Banach spaces.

When the theory of distributions was still not widely known nor used, property above was formulated by saying that the convolution of the function ' with a given function belonging to a proper Hilbert or Banach space converges as ε → 0 to this last one: this is exactly what Friedrichs did .< ref > See, properties PI, PII, PIII and their consequence PIII < sub > 0 </ sub >.</ ref > This also clarifies why mollifiers are related to approximate identities.

Its founders were Hugo Steinhaus and Stefan Banach, who were professors at the University of Lwów.

It was officially constituted on April 2, 1919. Hugo Steinhaus, Stefan Banach and Otto Nikodym were among the founders.