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algebra of sets – Banach space
algebra of sets
Given a set X, an algebra on X is a collection of subsets of X which is closed under finite unions and complements. Such an algebra always includes the empty set and X itself. An algebra may also be defined as a ring of sets which includes X. An algebra of sets which is closed under countable unions is called a s-algebra.
almost disjoint
Set Theory: If a is an infinite cardinal and x and y are subsets of a, then x and y are said to be almost disjoint of their intersection has cardinality less than a. An almost disjoint family is a subset A of the power set of a, such that any two distinct members of A of size a are almost disjoint. A maximal almost disjoint family is an almost disjoint family not properly contained in any other almost disjoint family. It is a theorem that if A is an almost disjoint family and |A| = a, then A is not maximal.
Cf. D-system.
almost everywhere
Given a measure m on a measure space X, a condition (e.g., continuity of functions, etc.) is said to hold almost everwhere if it holds on X – N, i.e., on the set difference of X and N, where N is a null set. This is often abbreviated by "a.e."
alternating group
The subgroup of a symmetric group on n elements which contains all of the even permutations is called the alternating group on n elements.
alternating series test
ARTICLE
A test for the convergence of a series. See the article for a complete description.
antichain
If X is a set with a strict partial order  , then a subset A of X is called an antichain if no x and y in A are related by .
Cf. chain.
antinomy
A paradox arising from formal mathematical axioms. An antinomy may lead to a strict contradiction, as in the Russell Paradox, or merely offend our intuition, as in the Banach-Tarski Paradox.
antisymmetric relation
A relation “~” on a set X is antisymmetric if, for all x and y in X, whenever x ~ y and y ~ x then x = y.
Cf. symmetric relation, asymmetric relation.
antitone function
A function that is order preserving and decreasing.
arithmetic
The mathematical theory of addition, subtraction, multiplication, and division of integers. Formally, arithmetic is usually axiomatized by the so-called Peano axioms, and the theory is then often referred to as PA (for “Peano arithmetic”).
Aronszajn tree
Set Theory: For a a regular cardinal, an a-Aronszajn tree is a tree T with |T| = a, such that every chain in T is of cardinality less than a. Every Suslin tree is an Aronszajn tree. There are no Aronszajn trees of height w.
Ascoli-Arzelŕ Theorem
If {X} is a compact Hausdorff space and F is an equicontinuous, pointwise bounded subset of the space C(X) of continuous functions on X, then F is totally bounded in the uniform metric and the closure of F in C(X) is compact. Also, if X is a s-compact, locally compact Hausdorff space and if (fn} is an equicontinuous, pointwise bounded sequence in C(X), then there exists a subsequence and an f in C(X) such that the subsequence converges to f uniformly on compact sets.
associative
An operation “ · ” on a set A is associative if for all a, b, and c in A, (a · b) · c = a · (b · c).
Cf. commutative, distributive.
associative property
A property of numbers which states that the operations of addition and multiplication are associative.
Cf. commutative, distributive.
asymmetric relation
A relation “~” on a set X is asymmetric if, for all x and y in X, x ~ y implies that it is not the case that y ~ x.
Cf. symmetric relation, antisymmetric relation.
automorphism
An isomorphism of a space to itself, that is, a bijective function on a space that preserves relationships, operations, and/or all essential properties (continuity, order, etc.).
Cf. homomorphism, group automorphism, ring automorphism, field automorphism.
automorphism group
The automorphisms of any object or structure can be composed to produce new automorphisms of that object. Since functional composition is associative, and since automorphisms are by their nature invertible, the automorphisms of an object or structure form a group (with the trivial automorphism serving as the identity element). This group is called the automorphism group of that object, usually denoted by Aut(X) for some object X. Automorphism groups are very useful, as they yield information about the symmetries of objects at hand.
axiom
In formal mathematics, a formula or schema of formulas stipulated as true in the theory under discussion, i.e., assumed to be true at the outset, and so not requiring proof. Axioms are the counterpart in mathematics of suppositions, assumptions, or premises in ordinary syllogistic logic.
axiom of choice
An axiom of set theory, particularly ZFC, which asserts that from any (infinite) family of sets a new set may be created containing exactly one element from each set in the family. This axiom has been shown to be independent of the other axioms of ZFC (ZF). It is equivalent to: 1) Zorn's Lemma; 2) the Hausdorff Maximality Theorem; 3) the well-ordering principle; and 4) the statement that the cartesian product of an infinite family of sets is non-empty. Because the axiom of choice permits non-constructive proofs, it is rejected by intuitionism, and careful mathematicians refer explicitly to its use in proofs which require it.
Baire Category Theorem
If X is a complete metric space, then the intersection of a countable collection of dense open subsets of X is also dense. Equivalently, the union of a countable collection of closed, nowhere dense subsets of X is nowhere dense.
Banach space
A complete normed space.
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