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Diametral Dimension of topological vector spaces

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Comprehensible Seminar

Diametral dimension of topological vector spaces

Loïc Demeulenaere

The concept of diametral dimension was introduced in 1961 by Bessaga, Pelczynski, Rolewicz and Mityagin as a topological invariant. Thus, it was used in a first time to prove that two topolo-gical vector spaces are not isomorphic. Then, a result due to Dynin and Mityagin showed that the diametral dimension can characterize two important types of functional spaces : Schwartz spaces and nuclear spaces. More recently (2010), Jean-Marie Aubry and Françoise Bastin determined the diametral dimension of Sν spaces and found that all these spaces have the same diametral

dimension (when they are locally p-convex).

In this presentation, the notion of topological invariants is introduced. Then, Kolmorov’s dia-meters are considered in order to define the diametral dimension of a topological vector space. After some properties, we study a practical example of spaces for which the diametral dimension is known (i.e. Köthe sequence spaces). Finally, we look at some properties of the diametral dimension in the context of Schwartz spaces and nuclear spaces and we consider the case of Sν spaces.

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