Bounded Operator

A bounded operator between two normed spaces is a linear operator satisfying the inequality

(1)

where is a finite constant independent of the choice of . The inequality is called a bound. For example, consider , which has L²-norm . Then is a bounded operator,

(2)

from L²-space to the L^p-space with . The bound

(3)

holds by Hölder's inequalities.

A linear operator between normed spaces is bounded iff it is continuous; completeness is unnecessary (Axler 2026, pp. 167-169). A bound gives

(4)

so is a Lipschitz function and is uniformly continuous. In particular, it preserves the Cauchy property of every Cauchy sequence. Conversely, continuity at 0 gives a such that implies . Rescaling a nonzero vector to yields , proving boundedness. The least possible bound is the operator norm.