A linear map between normed vector spaces that maps bounded sets to bounded sets; equivalently a linear operator that is continuous on the whole space and has finite operator norm.

Principle

Linearity together with a uniform bound on the operator's action on unit vectors yields continuity and control of outputs; boundedness is equivalent to continuity for linear maps between normed spaces.

Demonstration

On the Banach space ℓ2, the diagonal operator T defined by T((x_n))=(λ_n x_n) is bounded exactly when sup_n |λ_n|<∞; in finite-dimensional Euclidean spaces every linear map is bounded and its norm equals the matrix operator norm.

Misapplication

Treating an unbounded densely defined operator (such as differentiation on C[0,1] with the supremum norm) as bounded or assuming boundedness without checking the domain and norm; assuming every linear operator on an infinite-dimensional Banach space is bounded.

Consequence

When a linear operator is bounded one obtains an operator norm, continuity, applicability of the bounded inverse and closed graph theorems under hypotheses, and stability under norm limits of bounded operators.

Reversal

An unbounded linear operator fails to be continuous on the whole space; often it must be considered with a proper domain and may exhibit discontinuities or blow-up on sequences that are bounded in the domain space.

Boundary

Definition presumes normed vector spaces and maps defined on the whole space; it excludes densely defined unbounded operators, operators between topological vector spaces lacking norms, and pointwise-defined linear maps not respecting boundedness.

Semantic Tension

‘Bounded’ competes with ‘continuous’ (equivalent here), with the notion of boundedness on particular subsets versus global boundedness, and with the informal use of ‘bounded operator’ in contexts allowing unbounded domains; clarity about domain and topology resolves the tension.

Synthesis

A bounded linear operator is a globally continuous linear transformation between normed spaces characterized by a finite operator norm; this single property ties algebraic linearity to analytic control of output size, enabling standard functional-analytic theorems to apply.