, Banach space
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 φ1,φ2,...∈B* converges to φ∈B* in the weak-* topology iff for all x∈B, φi(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 Lp 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.