# 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

## 1 Answer

In functional analysis, boundedness of linear map is same as continuity of map.

As linear map is bounded invertible (so Surjective), then by open mapping theorem linear map is open map. Hence inverse of the map is continuous which is equivalent to boundedness of map.