Open-mapping theorem

From Encyclopedia of Mathematics
Revision as of 17:03, 7 February 2011 by (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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.


[1] K. Yosida, "Functional analysis" , Springer (1980)
[2] A.P. Robertson, W.S. Robertson, "Topological vector spaces" , Cambridge Univ. Press (1964)


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


[a1] M. de Wilde, "Closed graph theorems and webbed spaces" , Pitman (1978)
[a2] H.H. Schaefer, "Topological vector spaces" , Springer (1971)
[a3] H. Jarchow, "Locally convex spaces" , Teubner (1981) (Translated from German)
How to Cite This Entry:
Open-mapping theorem. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article