if it satisfies any of the following equivalent conditions: : For any neighborhood of the identity there exist finitely many such that For any neighborhood of there exists a finite subset such that (where the right hand side is the Minkowski sum ). For any neighborhood of there exist finitely many subsets of such that and each is -small. For any given filter subbase of the identity element's neighborhood filter (which consists of all neighborhoods of in ) and for every there exists a cover of by finitely many -small subsets of is : for every neighborhood of the identity and every countably infinite subset of there exist distinct such that (If is finite then this condition is satisfied vacuously). Any of the following three sets satisfies (any of the above definitions of) being (left) totally bounded: The closure of in This set being in the list means that the following characterization holds: is (left) totally bounded if and only if is (left) totally bounded (according to any of the defining conditions mentioned above). The same characterization holds for the other sets listed below. The image of under the canonical quotient which is defined by (where is the identity element). The sum The term usually appears in the context of Hausdorff topological vector spaces. In that case, the following conditions are also all equivalent to being (left) totally bounded: In the completion of the closure of is compact. Every ultrafilter on is a Cauchy filter. The definition of ''' is analogous: simply swap the order of the products. Condition 4 implies any subset of is totally bounded (in fact, compact; see above). If is not Hausdorff then, for example, is a compact complete set that is not closed. Topological vector spaces Any topological vector space is an abelian topological group under addition, so the above conditions apply. Historically, definition 1(b) was the first reformulation of total boundedness for topological vector spaces; it dates to a 1935 paper of John von Neumann. This definition has the appealing property that, in a locally convex space endowed with the weak topology, the precompact sets are exactly the bounded sets. For separable Banach spaces, there is a nice characterization of the precompact sets (in the norm topology) in terms of weakly convergent sequences of functionals: if is a separable Banach space, then is precompact if and only if every weakly convergent sequence of functionals