# Open-mapping theorem

A continuous linear operator $A$ mapping a Banach space $X$ onto all of a Banach space $Y$ is an open mapping, i.e. $A(G)$ is open in $Y$ for any $G$ which is open in $X$. This was proved by S. Banach. Furthermore, a continuous linear operator $A$ giving a one-to-one transformation of a Banach space $X$ onto a Banach space $Y$ is a homeomorphism, i.e. $A^{-1}$ 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 $X$ with values in $\mathbf R$ (in $\mathbf C$).

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

#### References

[1] | K. Yosida, "Functional analysis" , Springer (1980) |

[2] | A.P. Robertson, W.S. Robertson, "Topological vector spaces" , Cambridge Univ. Press (1964) |

#### Comments

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

#### References

[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: http://www.encyclopediaofmath.org/index.php?title=Open-mapping_theorem&oldid=33174