Encyclopedia > Open mapping theorem

  Article Content

Open mapping theorem

In mathematics, there are two theorems with the name open mapping theorem.

Functional analysis

In functional analysis, the open mapping theorem, also known as the Banach-Schauder theorem, is a fundamental result which states: if A : XY is a surjective continuous linear operator between Banach spaces X and Y, and U is an open set in X, then A(U) is open in Y.

The proof uses the Baire category theorem.

The open mapping theorem has two important consequences:

  • If A : XY is a bijective continuous linear operator between the Banach spaces X and Y, then the inverse operator A-1 : YX is continuous as well.
  • If A : XY is a linear operator between the Banach spaces X and Y, and if for every sequence (xn) in X with xn → 0 and Axny it follows that y = 0, then A is continuous (Closed graph theorem[?]).

Complex analysis

In complex analysis, the open mapping theorem states that if U is a connected open subset of the complex plane C and f : UC is a non-constant holomorphic function, then f(U) is an open subset of C.

All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article
Shinnecock Hills, New York

... age of 18 living with them, 52.8% are married couples living together, 6.8% have a female householder with no husband present, and 37.6% are non-families. 27.7% of all ...

This page was created in 24.2 ms