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 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.

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^*. 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) in R or C for all φ in X^* .

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.

The dual space X^* (if X is normed) 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^* .

[edit] 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

$\Phi_x:X^* \to \mathbb{R} \mbox{ or } \mathbb{C}$

defined by

$\Phi_x(\phi)=\phi(x) \mbox{ for all } \phi \in X^*$

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.

Furthermore, the unit ball of $X$ is compact in the weak topology if and only if $X$ is reflexive.

A sequence (or net) φ_n in $X *$ is convergent to φ in the weak* topology provided

$\phi_n (x) \rightarrow \phi (x) \mbox{ as } n \rightarrow \infty$ for all x in X.

[edit] References

Pedersen, Gert (1989). Analysis Now. Springer. ISBN 0-387-96788-5.
Willard, Stephen (February 2004). General Topology. Courier Dover Publications. ISBN 0-486-43479-6.