## Interesting inequality involving Banach spaces and an operator

Let be two Banach spaces with norms . Let (space of linear bounded operators ) be such that is closed and . Let denote another norm on which is weaker than , i.e. .

Prove that there exists a constant such that .

Haim Brezis,* Functional Analysis, Sobolev Spaces and Partial Differential Equations*, Chapter 2

**Proof: **Argue by contradiction. Suppose there exists a sequence of elements with and . We can assume that is surjective; otherwise, replace by . Then by the open mapping theorem, there exists such that . Therefore, there exists such that . Therefore with and . But is finite dimensional, and with it becomes a Banach space. On a Banach space two norms that are comparable are equivalent, and therefore are equivalent. We have that , which contradicts the fact that . Therefore, the assumption made is false, and the problem is solved.