Every totally bounded subset of a TVS is bounded. Every relatively compact set in a topological vector space is bounded. If the space is equipped with the weak topology the converse is also true. The set of points of a Cauchy sequence is bounded, the set of points of a Cauchy net need not be bounded. The closure of the origin (referring to the closure of the set ) is always a bounded closed vector subspace. This set is the unique largest (with respect to set inclusion ) bounded vector subspace of In particular, if is a bounded subset of then so is Unbounded sets A set that is not bounded is said to be unbounded. Any vector subspace of a TVS that is not a contained in the closure of is unbounded There exists a Fréchet space having a bounded subset and also a dense vector subspace such that is contained in the closure (in ) of any bounded subset of Stability properties In any TVS, finite unions, finite Minkowski sums, scalar multiples, translations, subsets, closures, interiors, and balanced hulls of bounded sets are again bounded. In any locally convex TVS, the convex hull (also called the convex envelope) of a bounded set is again bounded. However, this may be false if the space is not locally convex, as the (non-locally convex) Lp space spaces for have no nontrivial open convex subsets. The image of a bounded set under a continuous linear map is a bounded subset of the codomain. A subset of an arbitrary (Cartesian) product of TVSs is bounded if and only if its image under every coordinate projections is bounded. If and is a topological vector subspace of then is bounded in if and only if is bounded in In other words, a subset is bounded in if and only if it is bounded in every (or equivalently, in some) topological vector superspace of Properties A locally convex topological vector space has a bounded neighborhood of zero if and only if its topology can be defined by a seminorm. The polar of a bounded set is an absolutely convex and absorbing set. Using the definition of uniformly bounded sets given below, Mackey's countability condition can be restated as: If are bounded subsets of a metrizable locally convex space then there exists a sequence of positive real numbers such that are uniformly bounded. In words, given any