Weak topology

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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 or topological 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.

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

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The strong and weak topologies

Let X be a topological vector space, then in particular X is a topological space carrying a topology as part of its definition. (For example, a normed vector space X is, by using the norm to measure distances, also a topological vector space.) This topology is also called the strong topology on X.

The weak topology on X is defined using the continuous dual space 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 subbase for the weak topology is the collection of sets of the form φ-1(U) where φ ∈ X* and U is an open subset of the base field R or C. In other words, 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 is an intersection of finitely many sets of the form φ-1(U).

More generally, if X is a vector space and F is any family of linear functionals on X (in the algebraic dual space), then the initial topology of X with respect to the family F, denoted by σ(X,F), is sometimes also called the weak topology with respect to F. If F=X* is the continuous dual space of X, then the weak topology with respect to F coincides with the weak topology defined above. The weak topology σ(X,F) is induced by the family of seminorms,