complete topological vector space

As topological vector spaces are uniform spaces, it is appropriate to discuss completeness. As with a uniform space, a topological vector space is *complete* if it has no holes: everything that should be there actually is there.

Where this gets interesting is in the question as to what *should* be there. To determine this, one has to have some method of discovering holes. This is usually done by means of Cauchy nets, but in a given application it may not be necessary that all holes be filled and this leads to weaker notions of completeness.

The more general notions make sense for arbitrary topological vector spaces but some more refined notions are only used for locally convex topological vector spaces.

In strict order of decreasing strength, we have the following notions of completeness.

A locally convex topological vector space is **weakly complete** if it is complete for its weak topology.

A locally convex topological vector space is **$B$-complete** or is a **Ptak space** if a subspace of its continuous dual? is weakly closed? whenever its intersection with any equicontinuous subset is weakly closed (in the subset).

A locally convex topological vector space is **$B_r$-complete** if a dense subspace of its continuous dual? is weakly closed? whenever its intersection with any equicontinuous subset is weakly closed (in the subset).

A topological vector space is **complete** if every Cauchy net converges.

A topological vector space is **quasi-complete** if every bounded, closed? subset is complete.

A topological vector space is **sequentially complete** or **semi-complete** if every Cauchy sequence converges.

A locally convex topological vector space is **locally complete** if for $B \subseteq E$ a bounded?, closed?, absolutely convex subset then its norm space, $E_B$, is a Banach space.

Sequentially complete implies locally complete because every locally Cauchy sequence is a Cauchy sequence. The inverse implication “locally complete” $\Rightarrow$ sequentially complete is true for example in metrizable locally convex topological vector spaces, but not in general: A Cauchy sequence will not be locally Cauchy in general.

The problem to precisly characterize the spaces in which every convergent sequence is locally convergent is an open problem according to Köthe volume 1 (which is quite old, so it could have been solved in the meantime).

See functional analysis bibliography

Revised on May 21, 2010 05:41:28
by David Roberts
(203.24.207.116)