the topology below it (i.e. the topology generated by -complete and bounded disks) or if is not Hausdorff. If more than one collection of subsets appears the same row in the left-most column then that means that the same polar topology is generated by these collections. Notation: If denotes a polar topology on then endowed with this topology will be denoted by or simply For example, if then so that and all denote with endowed with Weak topology σ(Y, X) For any a basic -neighborhood of in is a set of the form: for some real and some finite set of points in The continuous dual space of is where more precisely, this means that a linear functional on belongs to this continuous dual space if and only if there exists some such that for all The weak topology is the coarsest TVS topology on for which this is true. In general, the convex balanced hull of a -compact subset of need not be -compact. If and are vector spaces over the complex numbers (which implies that is complex valued) then let and denote these spaces when they are considered as vector spaces over the real numbers Let denote the real part of and observe that is a pairing. The weak topology on is identical to the weak topology This ultimately stems from the fact that for any complex-valued linear functional on with real part then for all Mackey topology τ(Y, X) The continuous dual space of is (in the exact same way as was described for the weak topology). Moreover, the Mackey topology is the finest locally convex topology on for which this is true, which is what makes this topology important. Since in general, the convex balanced hull of a -compact subset of need not be -compact, the Mackey topology may be strictly coarser than the topology Since every -compact set is -bounded, the Mackey topology is coarser than the strong topology Strong topology 𝛽(Y, X) A neighborhood basis (not just a subbasis) at the origin for the topology is: The strong topology is finer than the Mackey topology. Polar topologies and topological vector spaces Throughout this section, will be a topological vector space (TVS) with continuous dual space and will be the canonical pairing, where by definition The vector space always distinguishes/separates the points of but may fail to distinguishes the points of (this necessarily happens if,