nLab
DF space

Contents

Idea

A DF space is a type of locally convex topological vector space. The basic idea is that a DF space is morally the strong dual of a Fréchet space. That is, it has all the nice structure that such a dual would have but without the bother of actually having to be a dual space.

Definition

Definition

A locally convex topological vector space is a DF space if it possesses a fundamental sequence? of bounded sets and if every strongly bounded countable union of equicontinuous subsets of the dual is again equicontinuous.

Properties

  1. The strong dual of a metrisable locally convex topological vector space is a DF space.
  2. Every normable space is a DF space.
  3. Every infrabarrelled space that has a fundamental sequence? of bounded sets is a DF space.
  4. In particular, every LB space? is a DF space, since any bounded set in an LB space is contained in one of the factors.
Proposition

A locally convex topological vector space that is metrisable and is a DF space is normable.

Proof

Let EE be a metrisable DF space.

To prove the result, we shall use a proposition recorded in Schaefer (IV.6.7). That says that a convex, circled subset VV of EE is a neighbourhood of 00 if (and only if) for every convex, circled bounded subset BEB \subseteq E, BVB \cap V is a 00-neighbourhood in BB.

As EE is metrisable, it has a countable 00-neighbourhood base, say (V n)(V_n). As EE is a DF space, it has a countable fundamental family of bounded sets, say (B n)(B_n). Let us assume, without loss of generality, that this family is increasing.

For each kk \in \mathbb{N}, as B kB_k is bounded, there is some λ k>0\lambda_k \gt 0 such that B kλ kV kB_k \subseteq \lambda_k V_k. Since, by assumption, the B kB_k are an increasing family, we have B kλ lV lB_k \subseteq \lambda_l V_l for all lkl \ge k. Let Bλ kV kB \coloneqq \bigcap \lambda_k V_k. This is a bounded set since it is contained in λ kV k\lambda_k V_k for each kk. Let ll \in \mathbb{N} and consider B lBB_l \cap B. We can write this as

B l( k=1 l1λ kV k)B l( klλ kV k). B_l \cap \big(\bigcap_{k=1}^{l-1} \lambda_k V_k\big) \cap B_l \cap \big( \bigcap_{k \ge l} \lambda_k V_k \big).

Since B lλ kV kB_l \subseteq \lambda_k V_k for klk \ge l, the last part is unnecessary and so we see that

B lB=B l( k=1 l1λ kV k). B_l \cap B = B_l \cap \big(\bigcap_{k=1}^{l-1} \lambda_k V_k \big).

This is then a finite intersection of open sets in B lB_l and so is open in B lB_l.

As this holds for any of the B lB_ls, it holds for any bounded set, whence the proposition from Schaefer applies to show that BB is a 00-neighbourhood. Thus EE possesses a bounded 00-neighbourhood, whence is normable.

References

See functional analysis bibliography.

Revised on May 21, 2010 10:53:25 by Andrew Stacey (129.241.15.70)