Bounded operator

The concept of a bounded linear operator has been extended from normed spaces to certain to all topological vector spaces.

A linear operator between normed spaces is bounded if and only if it is continuous.

Every sequentially continuous linear operator between TVS is a bounded operator.[1] This implies that every continuous linear operator is bounded. However, in general, a bounded linear operator between two TVSs need not be continuous.

This formulation allows one to define bounded operators between general topological vector spaces as an operator which takes bounded sets to bounded sets. In this context, it is still true that every continuous map is bounded, however the converse fails; a bounded operator need not be continuous. This also means that boundedness is no longer equivalent to Lipschitz continuity in this context.

If the domain is a bornological space (for example, a pseudometrizable TVS, a Fréchet space, a normed space) then a linear operators into any other locally convex spaces is bounded if and only if it is continuous. For LF spaces, a weaker converse holds; any bounded linear map from an LF space is sequentially continuous.