A set D in a Banach space E is limited if lim sup_(k→∞) sup_(z∈D) |#phi#_k(z)| > 0 =>sup_(||z||=1) lim sup_(k→∞) |#phi#_k(z)| > 0 for every sequence (#phi#_k) is contained in E~★. It is studied how this implication can be quantified, for example if there exists a constant C > 0 such that lim sup_(k→∞) sup_(z∈D) |#phi#_k(z)| = 1 => sup_(||z||=1) lim sup_(k→∞)|#phi#_k(z)| ≥ C for every sequence (#phi#_k(z) is contained in E~★, is studied. Relatively compact sets and limited sets in l~∞ - 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.
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