Two versions of diametral dimension are shown to coincide for quasinormable Frechet spaces. The diametral dimension is determined by a single bounded subset in certain cases. | Turkish Journal of Mathematics Research Article Turk J Math (2013) 37: 847 – 851 ¨ ITAK ˙ c TUB ⃝ doi: Quasinormability and diametral dimension ∗ ˙ GLU ˘ A. Tosun TERZIO Sabanc˘ g University, Faculty of Engineering and Natural Sciences ˙ 34956 Orhanl˘ g, Tuzla, Istanbul, Turkey Received: • Accepted: • Published Online: • Printed: Abstract: Two versions of diametral dimension are shown to coincide for quasinormable Fr´echet spaces. The diametral dimension is determined by a single bounded subset in certain cases. Key words and phrases: Diametral dimension, Fr´echet spaces, K¨ othe spaces 1. Introduction The set ∆(E) of sequences (ξn ) such that for each neighborhood U of zero of a locally convex space E there is another such neighborhood with lim ξn dn (V, U ) = 0 , where dn (V, U ) is the n -th diameter of V with respect to U , is called the diametral dimension of E . ([3], [6], [7], [8]). Another version is the set ∆b (E) of all sequences (ξn ) such that for each neighborhood U and each bounded subset B we have lim ξn dn (B, U ) = 0 . ∆b (E) is less frequently used than ∆(E). We always have c0 ⊂ ∆(E) ⊂ ∆b (E). In [6] Mitiagin claimed that ∆(E) = ∆b (E) holds for every Fr´echet space (F -space) E , referring for the proof to a forthcoming joint paper. However, there is an example of a K¨othe space λ(A), which is a Montel space but fails to be a Schwartz space. In this case we have ∆(λ(A)) = c0 ⊂ ℓ∞ ⊂ ∆b (λ(A)). On the other hand, if E is a locally convex space with a bounded subset that is not precompact, we have ∆(E) = ∆b (E) = c0 . We recall that a Fr´echet-Montel space (F M -space) is a Fr´echet-Schwartz space (F S -space) if and only if E is quasinormable [3]. There is an extensive literature concerning quasinormability (cf. [1]). We want to single out a remarkable result of Meise and Vogt [5], which states than an F -space is quasinormable if and .
