Weak topology

In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space (such as 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 concepts of functional analysis.

Topology where convergence of points is defined by the convergence of their image under continuous linear functionals
This article is about the weak topology on a normed vector space. For the weak topology induced by a general family of maps, see initial topology. For the weak topology generated by a cover of a space, see coherent topology.

One may call subsets of a topological vector space weakly closed (respectively, weakly compact, etc.) if they are closed (respectively, compact, etc.) with respect to the weak topology. Likewise, functions are sometimes called weakly continuous (respectively, weakly differentiable, weakly analytic, etc.) if they are continuous (respectively, differentiable, analytic, etc.) with respect to the weak topology.

. . . Weak topology . . .

Starting in the early 1900s, David Hilbert and Marcel Riesz made extensive use of weak convergence. The early pioneers of functional analysis did not elevate norm convergence above weak convergence and oftentimes viewed weak convergence as preferable.[1] In 1929, Banach introduced weak convergence for normed spaces and also introduced the analogous weak-* convergence.[1] The weak topology is also called topologie faible and schwache Topologie.

Let

K{displaystyle mathbb {K} }

be a topological field, namely a field with a topology such that addition, multiplication, and division are continuous. In most applications

K{displaystyle mathbb {K} }

will be either the field of complex numbers or the field of real numbers with the familiar topologies.

. . . Weak topology . . .

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. . . Weak topology . . .

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