not nowhere dense if and only if its closure is a neighborhood of the origin. A vector subspace of a TVS that is closed but not open is nowhere dense. Suppose is a TVS that does not carry the indiscrete topology. Then is a Baire space if and only if has no balanced absorbing nowhere dense subset. A TVS is a Baire space if and only if is nonmeager, which happens if and only if there does not exist a nowhere dense set such that Every nonmeager locally convex TVS is a barrelled space.Important algebraic facts and common misconceptionsIf then ; if is convex then equality holds. For an example where equality does hold, let be non-zero and set also works. A subset is convex if and only if for all positive real or equivalently, if and only if for all The convex balanced hull of a set is equal to the convex hull of the balanced hull of that is, it is equal to But in general, where the inclusion might be strict since the balanced hull of a convex set need not be convex (counter-examples exist even in ). If and is a scalar then If are convex non-empty disjoint sets and then or In any non-trivial vector space there exist two disjoint non-empty convex subsets whose union is Other properties' Every TVS topology can be generated by a of F-seminorms. If is some unary predicate (a true or false statement dependent on ) then for any So for example, if denotes "" then for any Similarly, if is a scalar then The elements of these sets must range over a vector space (that is, over ) rather than not just a subset or else these equalities are no longer guaranteed; similarly, must belong to this vector space (that is, ). Properties preserved by set operators The balanced hull of a compact (respectively, totally bounded, open) set has that same property. The (Minkowski) sum of two compact (respectively, bounded, balanced, convex) sets has that same property. But the sum of two closed sets need be closed. The convex hull of a balanced (resp. open) set is balanced (respectively, open). However, the convex hull of a closed set need be closed. And the convex hull of a bounded set need be bounded. The following table, the color of each cell indicates whether or not a given property of subsets