Bounded Operator

Bounded Operator

In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded by the same number, over all non-zero vectors v in X. In other words, there exists some M > 0 such that for all v in X

The smallest such M is called the operator norm of L.

A bounded linear operator is generally not a bounded function; the latter would require that the norm of L(v) be bounded for all v, which is not possible unless Y is the zero vector space. Rather, a bounded linear operator is a locally bounded function.

A linear operator on a metrizable vector space is bounded if and only if it is continuous.