# state open mapping theorem and use it to prove that the inverse of an invertible bounded linear map from Banach space X to Banach space Y is bounded

Use open mapping theorem to prove that the inverse of an invertible bounded linear map from Banach space X to Banach space Y is bounded.

I need simple implications of known theorems here to prove this.

Open mapping theorem: If X and Y are Banach spaces and A is a continuous linear surjection then A(G) is open in Y whenever G is open in X.

• This is immediate from the theorem and the definition of "continuous" in terms of open sets. – David C. Ullrich Nov 23 '17 at 17:58