On the equality between two diametral dimensions

Bastin, Françoise; Demeulenaere, Loïc

doi:10.7169/facm/1594

Cited by 2 publications

(1 citation statement)

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“…More information about diametral dimensions of Köthe spaces can be found in [Ter08,BD16]. A characterization of (Ω) for Köthe spaces is well-known, e.g., [Wag80, Satz1.10], and in fact easily obtained from π n * ℓ = 1/a ℓ (n) (where π n (x) = x n ) and the definition of (Ω):…”

Section: Prominent Setsmentioning

confidence: 99%

Diametral dimensions of Fréchet spaces

2016

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Abstract. The diametral dimension is an important topological invariant in the category of Fréchet spaces which has been used, e.g., to distinguish types of Stein manifolds. We introduce variants of the classical definition in order to solve an old conjecture of Bessaga, Mityagin, Pe lczyński, and Rolewicz at least for nuclear Fréchet spaces. Moreover, we clarify the relation between an invariant recently introduced by Terzioǧlu and the by now classical condition (Ω) of Vogt and Wagner. Kolmogorov widths and diametral dimensionsKolmogov widths (or diameters) are a quantitative measure for compactness in normed spaces: for absolutely convex sets V and U of a vector space X (typically U is the unit ball of a given norm) and n ∈ N 0 the n-th width is(the dependence on X is notationally surpressed). If V is bounded with respect to U (i.e., δ 0 (V, U ) < ∞) then V is precompact with respect to the the Minkowski functional of U (which is a seminorm with unit ball U ) if and only if δ n (V, U ) → 0. This elementary fact is proposition 1.2 in Pinkus' book [Pin85] where much more information about n-widths can be found.For a locally convex space (l.c.s) X with the system U 0 (X) of absolutely convex 0-neighbourhoods the diametral dimension of X is the sequence spaceThis space is a topological invariant, i.e., if X and Y are isomorphic l.c.s. thenMoreover, X is a Schwartz space (i.e., every 0-neighbourhood U contains another one which is precompact with respect to U ) if and only if ℓ ∞ ⊆ ∆(X). There are several versions of diametral dimensions of locally convex spaces in the literature, all going back to an idea of Pe lczyński [Pe l57] from 1957. The formulation above can be found in [Mit61] where Mityagin refers to a joint work with Bessaga, Pe lczyński, and Rolewicz which, to our best knowledge, eventually did not appear in print.

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Section: Prominent Setsmentioning

confidence: 99%