We give, departing from Grothendieck's description of the dual of the space of weak *-weak continuous finite-rank operators, a clear proof for the principle of local reflexivity in a general form.

Let L be an infinite locally compact Hausdorff topological space. We show that extremely regular subspaces of C0(L) have very strong diameter 2 properties and, for every real number ε with 0 < ε < 1, contain an ε-isometric copy of c0. If L does not contain isolated points they even have the Daugavet property, and thus contain an asymptotically isometric copy of ℓ1.2010 Mathematics Subject Classification. 46B20; 46B22.

We investigate sufficient and necessary conditions for the space of bounded linear operators between two Banach spaces to be rough or average rough. Our main result is that L(X,Y) is δ‐average rough whenever X* is δ‐average rough and Y is alternatively octahedral. This allows us to give a unified improvement of two theorems by Becerra Guerrero, López‐Pérez, and Rueda Zoca [J. Math. Anal. Appl. 427 (2015)].

Thin and thick sets in normed spaces were defined and studied by M.I. Kadets and V.P. Fonf in 1983. In this paper, we give a new characterization of thick sets in terms of weak integrability of Banach space valued measurable functions. We also characterize thick sets in terms of boundedness of vector measures, and explain how this concept is related to the theory of barrelled spaces.  2005 Elsevier Inc. All rights reserved.

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