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On the equality between two diametral dimensions. (English) Zbl 1409.46006

Let \(E\) be a locally convex space. The diametral dimension of \(E\) is the set \[ \Delta(E)=\{\xi\in\mathbb C^{\mathbb N_0}: \forall V\in \mathcal{V}(E)\;\exists U\in \mathcal{V}(E)\;\text{such\;that}\;U\subset V \;\text{and }\;(\xi_n \delta_n(U,V))_{n\in\mathbb N_0}\in c_0 \}, \] where \(\mathcal{V}(E)\) is a basis of \(0\)-neighbourhoods in \(E\) and \(\delta_n(U,V)\) is the \(n\)th Kolmogorov diameter of \(U\) with respect to \(V\), i.e., \[ \delta_n(U,V)=\inf\{\delta>0: \exists F\in \mathcal{L}_n(E) \;\text{such\;that }\;U\subset \delta V+F\}, \] where \(\mathcal{L}_n(E)\) denotes the set of all linear subspaces of \(E\) with a dimension less or equal to \(n\).
T. Terzioglu [Collect. Math. 20, 49–99 (1969; Zbl 0175.41602)] introduced another definition of diametral dimension as follows: \[ \Delta_b(E)=\{\xi\in\mathbb C^{\mathbb N_0}: \forall V\in \mathcal{V}(E)\;\forall B\in \mathcal{B}(E)\;\text{such\;that}\;(\xi_n \delta_n(B,V))_{n\in\mathbb N_0}\in c_0 \}, \] where \(\mathcal{B}(E)\) denotes the set of all bounded subsets of \(E\).
Obviously, \(\Delta(E)\subseteq\Delta_b(E)\) and \(\Delta(E)=\Delta_b(E)\) in the case that \(E\) is a normed space. But it is known that, if \(E\) is a Fréchet-Montel space which is not Schwartz, then \(\Delta(E)=c_0\subsetneq\Delta_b(E)\). The question of the equality \(\Delta(E)=\Delta_b(E)\) in the case of non-normable locally convex spaces remains an open problem even in the case of Fréchet-Schwartz spaces \(E\).
In the paper under review, the authors give sufficient conditions for the equality \(\Delta(E)=\Delta_b(E)\). They apply these conditions to Köthe echelon spaces defined with a regular Köthe matrix, to Köthe echelon spaces with property \((\overline{\Omega})\), and to Köthe echelon spaces of type \(G_\infty\). Moreover, the authors construct examples of nuclear (hence Schwartz) non-metrizable locally convex spaces for which the equality fails.

MSC:

46A63 Topological invariants ((DN), (\(\Omega\)), etc.) for locally convex spaces
46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
46A45 Sequence spaces (including Köthe sequence spaces)

Citations:

Zbl 0175.41602
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References:

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