Proving Banach spaces inverse operator property

Question

Let $X,Y$ be Banach spaces and $T\colon X\to Y$ is a linear continuous bijection. I want to prove that $T^{-1}$ is also continuous then.

Can I use closed graph theorem? Any ideas?

Answer 1

You can use the open mapping theorem:

"Let $X,Y$ Banach spaces and $T: X \to Y$ a continuous linear surjective operator. Then $T$ is open."

Proof: It is clear that $T^{-1}$ is linear. Using the open mapping theorem you can deduce that $T$ is open. Hence pre-images from open sets respecting $T^{-1}$ are open. Hence $T^{-1}$ is continuous.

I hope it helps you :)

Answer 2

The open map theorem tells us that if $X$ and $Y$ are two Banach spaces and $T:X \rightarrow Y$ is linear and continuous then $T$ is an open map. Now, if $T:X \rightarrow Y$ is also a bijection we have that $T^{-1}$ is also continuous because it's true this result of general topology:

If $f:X \rightarrow Y$ is bijection between topological spaces, then the following conditions are equivalent

(i) $f^{-1}$ is continuous

(ii) $f$ is open map

(iii) $f$ is closed map