uniform boundedness of functions
A single function is bounded if its graph fits between two horizontal lines. A whole family is uniformly bounded if ONE pair of lines contains every member at once — there is a single ceiling no function in the collection ever pokes through, no matter how many functions there are.
Definition: a family F of functions on a set E is uniformly bounded if there exists a constant M (independent of both x and the choice of function) such that |f(x)| <= M for all x in E and all f in F. Equivalently, sup over f in F of ||f|| <= M in the sup norm. This is stronger than each f being individually bounded, since that would only give a separate bound M_f per function, with no common ceiling.
Uniform boundedness is one of the two hypotheses of the Arzelà–Ascoli theorem, where it prevents the functions from escaping vertically while equicontinuity prevents horizontal wildness. A related but deeper phenomenon, the uniform boundedness principle (Banach–Steinhaus) for linear operators, shows that pointwise bounds can sometimes upgrade automatically to a uniform bound — but that is a special structural miracle, not the general situation here.
On R, the family {sin(nx) : n = 1, 2, ...} is uniformly bounded with M = 1. But {n x : n = 1, 2, ...} on (0, 1) is NOT uniformly bounded: each n x is individually bounded by n on (0,1), yet no single M dominates them all since the bounds n grow without limit.
Individually bounded need not mean uniformly bounded.