** The Hahn – Banach theorem in functional analysis, allowing the extension of linear functionals

* 1874 – Reynaldo Hahn, Venezuelan composer and conductor ( d. 1947 )

Banach spaces are named after the Polish mathematician Stefan Banach who introduced them in 1920 – 1922 along with Hans Hahn and Eduard Helly .< ref >

As a consequence of the Hahn – Banach theorem, this map is injective, and isometric.

* Hahn – Exton q-Bessel function

In mathematics, the Hahn – Banach theorem is a central tool in functional analysis.

" Another version of Hahn – Banach theorem is known as Hahn – Banach separation theorem or the separating hyperplane theorem, and has numerous uses in convex geometry.

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.

The Hahn – Banach theorem states that if is a sublinear function, and is a linear functional on a linear subspace U ⊆ V which is dominated by on U,

Another version of Hahn – Banach theorem states that if V is a vector space over the scalar field K ( either the real numbers R or the complex numbers C ), if is a seminorm, and is a K-linear functional on a K-linear subspace U of V which is dominated by on U in absolute value,

This reveals the intimate connection between the Hahn – Banach theorem and convexity.

The Mizar project has completely formalized and automatically checked the proof of the Hahn – Banach theorem in the HAHNBAN file.

The theorem has several important consequences, some of which are also sometimes called " Hahn – Banach theorem ":

Another version of Hahn – Banach theorem is known as the Hahn – Banach separation theorem.

* 1926 – Carl Hahn, German businessman

* 1950 – James Hahn, American politician

* 1906 – Anna Marie Hahn, German-American serial killer ( d. 1938 )

A famous example of a theorem of this sort is the Hahn – Banach theorem.

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.

One example is the Banach – Tarski paradox which says that it is possible to decompose (" carve up ") the 3-dimensional solid unit ball into finitely many pieces and, using only rotations and translations, reassemble the pieces into two solid balls each with the same volume as the original.

For example, the Banach – Tarski paradox is neither provable nor disprovable from ZF alone: it is impossible to construct the required decomposition of the unit ball in ZF, but also impossible to prove there is no such decomposition.

Statements such as the Banach – Tarski paradox can be rephrased as conditional statements, for example, " If AC holds, the decomposition in the Banach – Tarski paradox exists.

** The Banach – Tarski paradox.

** The Banach – Alaoglu theorem about compactness of sets of functionals.

# redirect Banach – Tarski paradox

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

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

However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.

** The Nielsen – Schreier theorem, that every subgroup of a free group is free.

** The numbers and are not algebraic numbers ( see the Lindemann – Weierstrass theorem ); hence they are transcendental.

This is a result of Galois theory ( see Quintic equations and the Abel – Ruffini theorem ).

Transmission, Gregory Chaitin also presents this theorem in J. ACM – Chaitin's paper was submitted October 1966 and revised in December 1968, and cites both Solomonoff's and Kolmogorov's papers.

This began in his doctoral work leading to the Mordell – Weil theorem ( 1928, and shortly applied in Siegel's theorem on integral points ).

He had introduced the adele ring in the late 1930s, following Claude Chevalley's lead with the ideles, and given a proof of the Riemann – Roch theorem with them ( a version appeared in his Basic Number Theory in 1967 ).

His ' matrix divisor ' ( vector bundle avant la lettre ) Riemann – Roch theorem from 1938 was a very early anticipation of later ideas such as moduli spaces of bundles.

* Borel – Weil theorem

* De Rham – Weil theorem

* Mordell – Weil theorem

His first ( pre-IHÉS ) breakthrough in algebraic geometry was the Grothendieck – Hirzebruch – Riemann – Roch theorem, a far-reaching generalisation of the Hirzebruch – Riemann – Roch theorem proved algebraically ; in this context he also introduced K-theory.

In 1956, he applied the same thinking to the Riemann – Roch theorem, which had already recently been generalized to any dimension by Hirzebruch.

The binomial theorem also holds for two commuting elements of a Banach algebra.

Since a maximal ideal in A is closed, is a Banach algebra that is a field, and it follows from the Gelfand-Mazur theorem that there is a bijection between the set of all maximal ideals of A and the set Δ ( A ) of all nonzero homomorphisms from A to C. The set Δ ( A ) is called the " structure space " or " character space " of A, and its members " characters.

Moreover, the Banach fixed point theorem states that every contraction mapping on a nonempty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f ( x ), f ( f ( x )), f ( f ( f ( x ))), ... converges to the fixed point.

The Banach fixed point theorem states that a contraction mapping on a complete metric space admits a fixed point.

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

* Banach fixed point theorem