** 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 – Banach theorem

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 )

In mathematics, the Vitali – Hahn – Saks theorem, introduced by,, and, states that given μ < sub > n </ sub > for each integer n > 0, a countably additive function defined on a fixed sigma-algebra Σ, with values in a given Banach space B, such that

In functional analysis the name Banach functional is used for sublinear function, especially when formulating Hahn – Banach theorem.

* Mary Exton Gaston ( 1855 – 1956 ), first female physician in Somerville and a " major force in the borough's development ".

* The Hon Charles Trent-Garby – Clive Exton

About half the trains on the Avocet Line from to call at Exton on request – this means that passengers alighting here must tell the conductor that they wish to do so, and those waiting to join must signal clearly to the driver as the train approaches.

Clive Exton ( 11 April 1930 – 16 August 2007 ) was a British television and film screenwriter, sometime playwright, and former actor.

Returning to England in 1986, Exton found that the television business had radically changed through the rise of the independent producer, such as Brian Eastman, for whom he wrote most of the episodes ( 20 ) of Agatha Christie ’ s Poirot, with David Suchet ( 1989 – 2000 ), all of the episodes ( 23 ) of Jeeves and Wooster, with Hugh Laurie and Stephen Fry ( 1990 – 1993 ), and ten episodes of Rosemary & Thyme ( 2003 – 2006 ).

* John Harington, 1st Baron Harington of Exton ( 1539 – 1613 ), English politician

* John Harington, 2nd Baron Harington of Exton ( 1592 – 1614 )

* Rutland Website – Exton

* SEPTA – Exton Station

* Exton Amtrak-SEPTA Station ( USA Rail Guide – Train Web )

* John Harington, 1st Baron Harington of Exton 16 May 1607 – 23 August 1613

* John Harington, 2nd Baron Harington of Exton 8 October 1613 – 27 February 1614

Animals sacred to Apollo included wolves, dolphins, roe deer, swans, cicadas ( symbolizing music and song ), hawks, ravens, crows, snakes ( referencing Apollo's function as the god of prophecy ), mice and griffins, mythical eagle – lion hybrids of Eastern origin.

One common version, the two-argument Ackermann – Péter function, is defined as follows for nonnegative integers m and n:

In mathematics, the Borsuk – Ulam theorem, named after Stanisław Ulam and Karol Borsuk, states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point.

Chorbishops are not generally ordained bishops – they are not given the sacrament of Holy Orders in that degree – but function as assistants to the diocesan bishop with certain honorary privileges.

This expression is related to the development of Bessel functions in terms of the Bessel – Clifford function.

This orthogonality relation can then be used to extract the coefficients in the Fourier – Bessel series, where a function is expanded in the basis of the functions J < sub > α </ sub >( x u < sub > α, m </ sub >) for fixed α and varying m.

* Bessel – Clifford function

The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in formulas for the sum of powers of the first positive integers, in the Euler – Maclaurin formula, and in expressions for certain values of the Riemann zeta function.

* Chief Specialist Officer or CSO – VP-level corporate officer responsible for a specific function or area at corporate level

* Chief Tax Officer or CTO – high-level corporate officer responsible for the tax function ( compliance, accounting and planning ) within a company.

* Director – a manager of managers within an organization who is often responsible for a major business function and who sometimes reports to a Vice President.

As the basis set size is increased, the energy and wave function tend towards a limit called the Hartree – Fock limit.

The Hartree – Fock wave function is a single configuration or determinant.

A typical application is furnished by the Arzelà – Ascoli theorem and in particular the Peano existence theorem, in which one is able to conclude the existence of a function with some required properties as a limiting case of some more elementary construction.

Under the Curry – Howard correspondence, the existence of currying and uncurrying is equivalent to the logical theorem, as tuples ( product type ) corresponds to conjunction in logic, and function type corresponds to implication.

In simple terms, the Church – Turing thesis states that a function is algorithmically computable if and only if it is computable by a Turing machine.

Informally the Church – Turing thesis states that if some method ( algorithm ) exists to carry out a calculation, then the same calculation can also be carried out by a Turing machine ( as well as by a recursively definable function, and by a λ-function ).

To establish that a function is computable by Turing machine, it is usually considered sufficient to give an informal English description of how the function can be effectively computed, and then conclude " By the Church – Turing thesis " that the function is Turing computable ( equivalently partial recursive ).

* Materials chemistry – preparation, characterization, and understanding of substances with a useful function.

It is also known, especially among physicists, as the Lorentz distribution ( after Hendrik Lorentz ), Cauchy – Lorentz distribution, Lorentz ( ian ) function, or Breit – Wigner distribution.