than all other -topologies on when considering only 's whose sets are subsets of If is a bornological space (e.g. metrizable or LF-space) then is complete. If is a normed space then the strong dual topology on may be defined by the norm where If is a LF-space that is the inductive limit of the sequence of space (for ) then is a Fréchet space if and only if all are normable. If is a Montel space then has the Heine–Borel property (i.e. every closed and bounded subset of is compact in ) On bounded subsets of the strong and weak topologies coincide (and hence so do all other topologies finer than and coarser than ). Every weakly convergent sequence in is strongly convergent. Mackey topology By letting be the set of all convex balanced weakly compact subsets of will have the Mackey topology on induced by or the topology of uniform convergence on convex balanced weakly compact subsets of , which is denoted by and with this topology is denoted by This topology is finer than and hence finer than Polar topologies induced by subsets of the continuous dual space Throughout, will be a TVS over the field with continuous dual space and the canonical pairing will be associated with and The table below defines many of the most common polar topologies on Notation: If denotes a polar topology on then endowed with this topology will be denoted by or (e.g. for we'd have so that and both denote with endowed with ). The closure of an equicontinuous subset of is weak-* compact and equicontinuous and furthermore, the convex balanced hull of an equicontinuous subset is equicontinuous. Weak topology Suppose that and are Hausdorff locally convex spaces with metrizable and that is a linear map. Then is continuous if and only if is continuous. That is, is continuous when and carry their given topologies if and only if is continuous when and carry their weak topologies. Convergence on equicontinuous sets If was the set of all convex balanced weakly compact equicontinuous subsets of then the same topology would have been induced. If is locally convex and Hausdorff then 's given topology (i.e. the topology that started with) is exactly That is, for Hausdorff and locally convex, if then is equicontinuous if and only if is equicontinuous and furthermore, for any is a neighborhood of the origin if and only