# Is there a Baire Category Theorem for Complete Topological Vector Spaces?

Baire category theorem is usually proved in the setting of a complete metric space or a locally compact Hausdorff space.

Is there a version of Baire category Theorem for complete topological vector spaces? What other hypotheses might be required?

• I think this question might benefit from clarification about what "complete" means, vis-a-vis Cauchy nets. E.g., one can prove that weak duals of infinite-dimensional Hilbert spaces are not fully complete, in the sense that there are Cauchy nets that do not converge. Yet these weak duals are still quasi-complete, in the sense that (TVS-sense) bounded Cauchy nets converge, and this is enough for many purposes. (Quasi-complete is equivalent to completeness for metric spaces.) – paul garrett Mar 21 '17 at 21:43
• @paulgarrett Is it enough to have Hausdorffness to replace Cauchy nets with Cauchy sequences, and would then the two notions of completeness coincide? – Thangachelli Debopritama Mar 22 '17 at 4:49

## 2 Answers

The Baire theorem also holds for topologically complete spaces, i.e. Tychonov spaces that are a $G_\delta$ in some compactification;

IIRC this includes topological vector spaces that are complete in their uniformity as well.  This is actually false, see the answers for this question....

For large classes of TVS (like Fréchet spaces or Banach spaces) we can apply the metric version, and many weak topologies are (sequentially) complete as well.

More on completeness of TVS can be found here

According to Birkhoff-Kakutani theorem, a topological vector space is metrisable if and only if it is Hausdorff and $0$ has a countable neighborhood basis. (For a topological vector space, being Hausdorff is equivalent to $\{0\}$ being closed.)

Therefore, a complete topological vector space in which $\{0\}$ is closed and $0$ has a countable neighborhood basis satisfies the Baire category theorem.

• A complete topological vector space. You probably mean. – Henno Brandsma Mar 22 '17 at 4:22