convex space and is absorbing in then is bounded in Moreover, a subset is weakly bounded if and only if is absorbing in For this reason, it is common to restrict attention to families of bounded subsets of Weak/weak* topology The topology has the following properties: Banach–Alaoglu theorem: Every equicontinuous subset of is relatively compact for it follows that the -closure of the convex balanced hull of an equicontinuous subset of is equicontinuous and -compact. Theorem (S. Banach): Suppose that and are Fréchet spaces or that they are duals of reflexive Fréchet spaces and that is a continuous linear map. Then is surjective if and only if the transpose of is one-to-one and the image of is weakly closed in Suppose that and are Fréchet spaces, is a Hausdorff locally convex space and that is a separately-continuous bilinear map. Then is continuous. In particular, any separately continuous bilinear maps from the product of two duals of reflexive Fréchet spaces into a third one is continuous. is normable if and only if is finite-dimensional. When is infinite-dimensional the topology on is strictly coarser than the strong dual topology Suppose that is a locally convex Hausdorff space and that is its completion. If then is strictly finer than Any equicontinuous subset in the dual of a separable Hausdorff locally convex vector space is metrizable in the topology. If is locally convex then a subset is -bounded if and only if there exists a barrel in such that Compact-convex convergence If is a Fréchet space then the topologies Compact convergence If is a Fréchet space or a LF-space then is complete. Suppose that is a metrizable topological vector space and that If the intersection of with every equicontinuous subset of is weakly-open, then is open in Precompact convergence Banach–Alaoglu theorem: An equicontinuous subset has compact closure in the topology of uniform convergence on precompact sets. Furthermore, this topology on coincides with the topology. Mackey topology By letting be the set of all convex balanced weakly compact subsets of will have the Mackey topology on or the topology of uniform convergence on convex balanced weakly compact sets, which is denoted by and with this topology is denoted by Strong dual topology Due to the importance of this topology, the continuous dual space of is commonly denoted simply by Consequently, The topology has the following properties: If is locally convex, then this topology is finer