 Continuous linear operator

A continuous linear operator is a linear map that is continuous for the topologies of its domain and codomain. Between normed vector spaces, continuity is equivalent to an operator bound $\Vert Tx\Vert \le C\Vert x\Vert$ for some finite $C$, so the terms bounded linear operator and continuous linear operator coincide there.

For general topological vector spaces, sending every bounded set in a topological vector space to a bounded set is a weaker property. For example, the identity from the weak topology on an infinite-dimensional Hilbert space to its norm topology maps bounded sets to bounded sets by the Uniform boundedness principle, but is not continuous. A basic weak zero neighbourhood only restricts finitely many linear functionals, leaving a nonzero common-kernel direction with arbitrarily large norm.

Thus the operator norm interpretation of boundedness used throughout numerical linear algebra belongs to the normed-space setting. In other vector topologies, specify continuity explicitly.

A **continuous linear operator** is a specific type of mapping between two vector spaces that preserves both the structures of linearity and continuity.