In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a normed vector space with respect to its (continuous) dual. The remainder of this article will deal with this case, which is one of the basic concepts of functional analysis. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Jump to: navigation, search In topology and related areas of mathematics, the initial topology (projective topology or weak topology) on a set , with respect to a family of functions on , is the coarsest topology on X which makes those functions continuous. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ... Jump to: navigation, search In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ... Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ...

Every normed vector space X is, by using the norm to measure distances, a metric space and hence a topological space. This topology on X is also called the strong topology. The weak topology on X is defined using the continuous dual space X^* (X '). This dual space consists of all linear functions from X into the base field R or C which are continuous with respect to the strong topology. The weak topology on X is the weakest topology (the topology with the fewest open sets) such that all elements of X^* remain continuous. Explicitly, a subset of X is open in the weak topology if and only if it can be written as a union of (possibly infinitely many) sets, each of which being an intersection of finitely many sets of the form φ^-1(U) with φ in X^* and U an open subset of the base field R or C. A sequence (x_n) in X converges in the weak topology to the element x of X if and only if φ(x_n) converges to φ(x) for all φ in X^* . In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ... This is a page about mathematics. ...

If X is equipped with the weak topology, then addition and scalar multiplication remain continuous operations, and X is a locally convex topological vector space. In functional analysis, a topological vector space is called locally convex if its topology is defined by a set of convex neighborhoods of 0. ...

The dual space X^* is itself a normed vector space by using the norm ||φ|| = sup_||x||≤1|φ(x)|. This norm gives rise to the strong topology on X^* .

The weak* topology

One may also define a weak* topology on X ^* by requiring that it be the weakest topology such that for every x in X, the substitution map

defined by

remains continuous.

An important fact about the weak* topology is the Banach-Alaoglu theorem: the unit ball in X ^* is compact in the weak* topology. The Banach-Alaoglu theorem (also known as Alaoglus Theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the product of compact sets with the product topology. ... In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has...

Furthermore, the unit ball of X is compact in the weak topology if and only if X is reflexive. This page concerns the reflexivity of a Banach space. ...

See also

• Weak convergence (Hilbert space)
• Weak-star operator topology
• Topologies on the set of operators on a Hilbert space