nLab complete topological vector space

Idea

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.

Definitions

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

Definition

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

Definition

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).

Definition

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).

Definition

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

Definition

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

Definition

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

Definition

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.

Properties

Sequentially Complete versus Locally Complete

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).

References

See functional analysis bibliography