E -approximation properties
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E-approximation properties Ju Myung Kim1 · Bentuo Zheng2 Received: 8 April 2020 / Accepted: 25 August 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract In this paper, we introduce the E-approximation property and the E u -approximation property which generalize Sinha and Karn’s p-approximation property. Characterizations of these properties are given parallel to Lima and Oja’s results on the strong approximation property and the weak bounded approximation property. Representation theorems for the dual of L(X , Y ) under the topology of uniform convergence on E-compact sets and E u -compact sets are also provided. As an application, the representations are used to generalize the main theorem of Choi and Kim in [1]. These results build up fundamental bases for future investigations of these properties. Keywords Approximation property · Schauder basis · Unconditional basis Mathematics Subject Classification 46B28 · 47L20
1 Introduction One of the most important properties in Banach space theory is the approximation property (AP) which was systematically investigated by Grothendieck [3]. A Banach space X is said to have the AP if τc
id X ∈ F (X , X ) , where id X is the identity map on X , F (X , X ) is the space of all finite rank operators from X to X and τc is the topology of uniform convergence on compact sets, more precisely, for
Ju Myung Kim was supported by National Research Foundation of Korea (NRF-2018R1D1A1B07043566). Bentuo Zheng was supported in part by Simons Foundation Grant 585081.
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Bentuo Zheng [email protected] Ju Myung Kim [email protected]
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Department of Mathematics and Statistics, Sejong University, Seoul 05006, South Korea
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Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152-3240, USA 0123456789().: V,-vol
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every compact subset K of X and every > 0, there exists an S ∈ F (X , X ) such that sup x − Sx ≤ . x∈K
In the last decade, some variants of the AP have been intensively studied, which originate from Sinha and Karn’s p-approximation property ( p-AP) [9]. They was motivated by a Grothendieck’s criterion [3] of compactness as follows. A subset K of a Banach space X is relatively compact if and only if there exists a null sequence (xn )n ∈ c0 (X ) such that ∞ K ⊂ αn xn : (αn )n ∈ B1 , n=1
where we denote by B Z the unit ball of a Banach space Z . Let 1 ≤ p < ∞. A subset K of X is said to be p-compact [9] if there exists (xn )n ∈ p (X ) such that ∞ αn xn : (αn )n ∈ B p∗ , K ⊂ p-co(xn )n := n=1
1/ p ∗
= 1 and ( p (X ), · p ) is the Banach space of absolutely p-summable where 1/ p + sequences in X . A Banach space X is said to have the p-AP when the compact set in the definition of the AP is replaced by the p-compact set. The main purpose of this paper is to introduce more general types of approximation property called the E-approximation property and the E u -approximation property (see Sect. 2 for definitions) and to give characterizations of them. Representation theorems fo
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