uniform boundedness principle
The uniform boundedness principle turns a family of separate, pointwise observations into one global guarantee. If a whole collection of operators stays bounded at each individual vector — no single input ever produces an infinite output across the family — then in fact they share a single finite bound on their norms. Local control everywhere secretly amounts to uniform control.
Precisely: let X be a Banach space, Y a normed space, and { T_a } a family of bounded linear operators from X to Y. Suppose the family is pointwise bounded: for each x in X the set { ||T_a x|| } is bounded (the bound may depend on x). Then the family is uniformly bounded: there is a single M with ||T_a|| <= M for every a. The supremum of operator norms is finite.
Completeness of the domain X is the load-bearing hypothesis; the proof runs through the Baire category theorem, and the principle is false without it. A celebrated consequence: if a sequence T_n of bounded operators converges pointwise to a map T (that is, T_n x -> T x for every x), then the T_n are uniformly bounded, the limit T is itself a bounded linear operator, and ||T|| <= liminf ||T_n||. This is how one safely takes limits of operators.
On l^2 let T_n x = (x_1, 2 x_2, ..., n x_n, 0, 0, ...). For each fixed x the values ||T_n x|| are bounded (they stabilize once n exceeds the support of x), yet ||T_n|| = n -> infinity. There is no contradiction: l^2 is complete but no single x has its outputs bounded uniformly only because each x has finitely significant coordinates — the family really is pointwise bounded only on the dense subspace of finitely supported sequences, not all of l^2.
Multiplier operators with norms n; the principle forces pointwise boundedness to fail on some vector.
Read the contrapositive: if a family is NOT uniformly bounded, then some single vector witnesses the unboundedness — there exists an x at which the outputs blow up. This is the engine behind classic constructions of continuous functions whose Fourier series diverge at a point.