A continuous linear operator mapping a Banach space onto all of a Banach space is an open mapping, i.e. is open in for any which is open in . This was proved by S. Banach. Furthermore, a continuous linear operator giving a one-to-one transformation of a Banach space onto a Banach space is a homeomorphism, i.e. is also a continuous linear operator (Banach's homeomorphism theorem).

The conditions of the open-mapping theorem are satisfied, for example, by every non-zero continuous linear functional defined on a real (complex) Banach space with values in (in ).

The open-mapping theorem can be generalized as follows: A continuous linear operator mapping a fully-complete (or -complete) topological vector space onto a barrelled space is an open mapping. The closed-graph theorem can be considered alongside with the open-mapping theorem.

References

A recent comprehensive study of the closed-graph theorem can be found in [a1].

References