Contemporary developments in logic and the foundations of mathematics, especially Bertrand Russell and Alfred North Whitehead's monumental Principia Mathematica, impressed the more mathematically minded logical positivists such as Hans Hahn and Rudolf Carnap.

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 mathematics, especially in the area of abstract algebra dealing with ordered structures on abelian groups, the Hahn embedding theorem gives a simple description of all linearly ordered abelian groups.

The technique has been applied in the study of mathematics and logic since before Aristotle ( 384 – 322 B. C.

In mathematics, the axiom of regularity ( also known as the axiom of foundation ) is one of the axioms of Zermelo – Fraenkel set theory and was introduced by.

The axiom of regularity is arguably the least useful ingredient of Zermelo – Fraenkel set theory, since virtually all results in the branches of mathematics based on set theory hold even in the absence of regularity ( see chapter 3 of ).

In mathematics, the arithmetic – geometric mean ( AGM ) of two positive real numbers and is defined as follows:

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.

His father, Étienne Pascal ( 1588 – 1651 ), who also had an interest in science and mathematics, was a local judge and member of the " Noblesse de Robe ".

William Frederick Schelter ( 1947 – July 30, 2001 ) was a professor of mathematics at The University of Texas at Austin and a Lisp developer and programmer.

Similarly, the influences of philosophers such as Sir Francis Bacon ( 1561 – 1626 ) and René Descartes ( 1596 – 1650 ), who demanded more rigor in mathematics and in removing bias from scientific observations, led to a scientific revolution.

The contributions of Kurt Gödel in 1940 and Paul Cohen in 1963 showed that the hypothesis can neither be disproved nor be proved using the axioms of Zermelo – Fraenkel set theory, the standard foundation of modern mathematics, provided ZF set theory is consistent.

He passed the examination in the elements of mathematics and the theory of navigation at the Royal Naval Academy on 2 – 4 September 1816, and became a 1st Lieutenant on 1 September 1818.

In 1949, while doing unrelated archival work, the historian of mathematics Carolyn Eisele ( 1902 – 2000 ) chanced on an autograph letter by Peirce.

* Theoretical chemistry – study of chemistry via fundamental theoretical reasoning ( usually within mathematics or physics ).

In mathematics, the Cauchy – Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which must be satisfied if we know that a complex function is complex differentiable.

" The new grounding of mathematics: First report ," 1115 – 33.

" The logical foundations of mathematics ," 1134 – 47.

" The foundations of mathematics ," with comment by Weyl and Appendix by Bernays, 464 – 89.

In contrast to real numbers that have the property of varying " smoothly ", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values.

The Englert – Greenberger duality relation provides a detailed treatment of the mathematics of double-slit interference in the context of quantum mechanics.

* 500 – Science ( including mathematics )

In mathematics, an infinite series will sometimes converge –

In mathematics, the Euler – Maclaurin formula provides a powerful connection between integrals ( see calculus ) and sums.

In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain ( also called a Euclidean ring ) is a ring that can be endowed with a certain structure – namely a Euclidean function, to be described in detail below – which allows a suitable generalization of the Euclidean division of the integers.

In mathematics, more specifically in functional analysis, a Banach space ( pronounced ) is a complete normed vector space.

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.

* The Banach space of functions of bounded variation is not separable ; note however that this space has very important applications in mathematics, physics and engineering.

Born in Kraków, Banach enrolled in " Henryk Sienkiewicz Gymnasium " and worked on mathematics problems with his friend Witold Wiłkosz.

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.

Later in life Banach would credit Dr. Kamil Kraft, the mathematics and physics teacher at the gymnasium with kindling his interests in mathematics.

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

In mathematics, the Banach fixed-point theorem ( also known as the contraction mapping theorem or contraction mapping principle ) is an important tool in the theory of metric spaces ; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points.

A key theme from the " categorical " point of view is that mathematics requires not only certain kinds of objects ( Lie groups, Banach spaces, etc.

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

* Spectral theory, in mathematics, a theory that extends eigenvalues and eigenvectors to linear operators on Hilbert space, and more generally to the elements of a Banach algebra

In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra ( or more generally a Banach algebra ), such that representations of the algebra are related to representations of the group.

In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the weakest locally convex topology on the set of bounded operators on a Hilbert space ( or, more generally, on a Banach space ) such that the evaluation map sending an operator T to the real number is continuous for each vector x in the Hilbert space.

In mathematics, the closed graph theorem is a basic result in functional analysis which characterizes continuous linear operators between Banach spaces in terms of the operator graph.

In functional analysis and related branches of mathematics, the Banach – Alaoglu theorem ( also known as Alaoglu's theorem ) states that the closed unit ball of the dual space of a normed vector space is compact in the weak * topology.

In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y.

In mathematics, Tsirelson space T is an example of a reflexive Banach space in which neither an l < sup > p </ sup > space nor a c < sub > 0 </ sub > space can be embedded.

In mathematics, a Banach space is said to have the approximation property ( AP ), if every compact operator is a limit of finite-rank operators.

In mathematics, the ba space of an algebra of sets is the Banach space consisting of all bounded and finitely additive measures on.

In mathematics, a polynomially reflexive space is a Banach space X, on which the space of all polynomials in each degree is a reflexive space.