Uniform Bounds for Limited Sets and Applications to Bounding Sets.

Abstract

A set D in a Banach space E is limited if lim sup k3I sup zPD j9 k zj b 0 Asup kzk1 lim sup k3I j9 k zjb 0 for every sequence 9 k & E c. It is studied how this implication can be quantified, for example if there exists a constant C b 0 such that lim sup k3I sup zPD j9 k zj 1 A sup kzk1 lim sup k3I j9 k zj ! C for every sequence 9 k & E c , is studied. Relatively compact sets and limited sets in l I-among others the unit vectors-have uniform bounds in this sense. A fundamental example of a limited set without any uniform bounds is constructed. A set D is called bounding if f D is bounded for every entire function on E. That bounding sets are uniformly limited and that strongly bounding sets are limited in the strongest sense are proved. Examples show that the convex hull of bounding sets in general are not bounding and that bounding sets in general does not have Grothendieck's incapsulating property as relatively weakly compact sets have. lim k3I sup zPD j9 k zj 0 An equivalent formulation without referring to null sequences is: D is called limited if lim sup k3I sup zPD j9 k zj b 0 A sup kzk1 lim sup k3I j9 k zj b 0 for every sequence 9 k & E c. A natural question is how this relation can be quantified. For example we may ask if a limited set D has a constant C b 0 such that lim sup k3I sup zPD j9 k zj 1 A sup kzk1 lim sup k3I j9 k zj ! C for every sequence 9 k & E c. Such a limited set is called limited in the