A functional-analytic result (Banach–Steinhaus theorem) stating that a family of continuous linear operators on a Banach space that is pointwise bounded is uniformly bounded in operator norm: pointwise boundedness implies a global supremum bound.

Principle

Use Baire category on the Banach domain to promote pointwise control (for each x, sup_T ||T x|| < ∞) to uniform control (sup_T ||T|| < ∞), giving a rule that local boundedness forces global boundedness under completeness.

Demonstration

Let X be a Banach space and F a family of bounded linear maps X→Y with sup_{T∈F} ||T x|| < ∞ for every x; Baire's theorem produces a nonempty open set on which the sup over F of ||T|| is finite, hence a uniform operator norm bound follows.

Misapplication

Assuming the conclusion without the domain being complete (Banach) is a common misuse; there are pointwise-bounded families on incomplete spaces whose operator norms are unbounded.

Consequence

Prevents pathological examples where pointwise boundedness coexists with arbitrarily large operator norms, underpins many convergence and compactness arguments and justifies interchanging limits in functional settings.

Reversal

Dropping completeness or linearity reverses the conclusion: in incomplete spaces or for nonlinear maps, pointwise boundedness need not imply any uniform bound, and counterexamples are hence common.

Boundary

Requires a Banach domain and linear continuous operators; the target may be only a normed space; the theorem does not directly apply to nonlinear operators, pointwise convergence of functions, or families lacking measurability hypotheses where required.

Semantic Tension

Tension arises with pointwise notions like pointwise convergence: pointwise boundedness is weaker and may coexist with divergent operator norms absent the Banach hypothesis; the principle elevates pointwise to uniform control only in the complete linear setting.

Synthesis

The Principle of Uniform Boundedness asserts that in a complete linear setting, local boundedness at each point forces a global operator-norm bound: completeness plus linearity transforms pointwise finiteness of images into a uniform supremum on operator norms, preventing hidden unboundedness.