Theorem of the open function

In mathematics, there are two theorems with the name "Open function theorem." Functional analysis In function analysis, the open-function theorem, also known as Banach-Schauder's theorem, is a fundamental result that states: if A: X → Y is a continuous linear operator between the Banach spaces X and y And, then A is an open function (ie if U is an open set in X, then A (U) is open in Y).

The test uses Baire's category theorem.

The open function theorem has two important consequences: Complex Analysis In complex analysis, the open function theorem states that if U is a connected open subset of the complex plane C and f: U → C is a non-constant holomorphic function, then f is an open function (ie it sends subsets open U's to the open subsets of C).

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