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Functional analysis,

Banach spaces:)

Let Banach space B^{*} be the dual of Banach space B: B^{*} is the set of *continuous* linear functionals on B. **NOTE** that it is possible that B^{*}=C^{*} for non-isomorphic Banach spaces B and C, which is a nuisance. So the weak-* topology depends also on B, not just on B^{*}. The *weak-* topology* (or *weak dual topology*, although that name is never used) is a topology conceptually similar to the weak topology, only for convergence of linear functionals.

Like the weak topology, it is simplest to specify it in terms of a test for convergence (it is true, albeit not obvious, that this test really does specify one single topology). A sequence of linear functionals φ_{1},φ_{2},...∈B^{*} converges to φ∈B^{*} in the weak-* topology iff for all x∈B, φ_{i}(x)->φ(x).

This is very similar to the weak topology, but reversing the rôles of B and B^{*}. Like the weak topology, the weak-* topology is *weaker* than the topology on B^{*} induced by the metric of the norm of B^{*}. If B is a reflexive Banach space (e.g. any Banach space L^{p} with 1<p<∞, and especially any Hilbert space) then (B^{*})^{*} has a natural isomorphism to B, and the weak-* topology is "just" the weak topology. In particular, the example given in weak topology of a sequence in Hilbert space which converges weakly but not in norm is also an example of a sequence which converges weak-*-ly but not in norm.

In any case we have a natural inclusion of B in (B^{*})^{*}, which shows that convergence in the weak topology implies convergence in the weak-*-* topology. This topology, however, is even weaker than the weak topology.

The Alaoglu theorem (no, really) is an important result about the weak-* topology.