bounded linear operator
A bounded linear operator is a linear map between normed spaces that never blows distances up by more than a fixed factor. Think of it as a machine that takes vectors in and puts vectors out, respecting addition and scaling, and which has a guaranteed maximum amplification: no input gets stretched by more than some finite number times its length.
Precisely, let X and Y be normed spaces and T : X -> Y linear, meaning T(a x + b y) = a T x + b T y. We call T bounded if there is a constant M with ||T x|| <= M ||x|| for all x in X. The smallest such M is the operator norm ||T||. The key fact is that for linear maps boundedness is exactly the same as continuity: a linear T is continuous if and only if it is bounded, and then it is even uniformly continuous and Lipschitz.
Beware the terminology: “bounded” here does NOT mean the image is a bounded set — a nonzero linear map sends the whole unbounded space onto an unbounded image. It means bounded on the unit ball, i.e. a uniform bound on the stretch factor. In infinite dimensions unbounded (discontinuous) linear maps genuinely exist, such as differentiation on the space of polynomials, so boundedness is a real hypothesis, not an automatic one.
Integration T f = integral from 0 to 1 of f is a bounded functional on C[0, 1]: |T f| <= integral |f| <= max|f| = ||f||, so ||T|| <= 1 (in fact = 1, attained by f = 1). Differentiation D f = f', however, is unbounded: take f_n(x) = sin(n x)/n with ||f_n|| = 1/n -> 0 yet ||D f_n|| = ||cos(n x)|| = 1, so no finite M works.
Integration is bounded; differentiation is the classic unbounded operator.
Why are boundedness and continuity the same only for LINEAR maps? Linearity lets you transport a single bound at the origin everywhere: if T is continuous at 0 it is continuous everywhere, and continuity at 0 means small inputs give small outputs, which is exactly a stretch bound. Nonlinear maps lack this homogeneity, so for them the two notions diverge.