Complete set

in a topological vector space over a field

A set such that the set of linear combinations of the elements from is (everywhere) dense in , i.e. the closed subspace generated by the set , i.e. the closed linear hull of , coincides with . For example, in the normed space of continuous functions on with values in the set is a complete set. If is a non-discretely normed field, each absorbing set (and in particular each neighbourhood of zero in ) is a complete set.

In order for , , to be a complete set in the weak topology of a space it is necessary and sufficient that there exist an index such that for each ; this means that no closed hyperplane contains all the elements , i.e. that is a total set. Moreover, if is a locally convex space, a complete set in the weak topology will be a complete set in the initial topology also.

Comments

Of course, a complete set in a topological vector space can also mean a set such that every Cauchy sequence in converges in ; and this is by far the most frequently occurring meaning of the phrase. For the notion of an absorbing set cf. Topological vector space.