Uniform Bounds for Limited Sets and Applications to Bounding Sets.

Abstract

A set D in a Banach space E is limited if lim sup(k-->infinity) sup(z epsilon D)\phi k(z) > 0 double right arrow sup(\\z\\=1) lim sup(k-->infinity) \phi k(z) > 0 for every sequence (phi k) subset of E*. It is studied how this implication can be quantified, for example if there exists a constant C > 0 such that lim sup(k-->infinity)sup(z epsilon D)\phi k(z) = 1 double right arrow sup\\z = 1 lim sup(k-->infinity) \phi k(z) greater than or equal to C for every sequence (phi k) subset of E*, is studied. Relatively compact sets and limited sets in l(infinity) - 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.