MHB Linear operator, its dual, proving surjectivity

[Linux]

Let $$T: X \rightarrow Y$$ be a continuous linear operator between Banach spaces.

Prove that $T$ is surjective $$\iff$$ $$T^*$$ is injective and $$im T^*$$ is closed.

I've proven a "similar" statement, with $$imT^*$$ replaced with $$imT$$.

There I used these facts: $\overline{imT}= ^{\perp}(kerT^*)$ and $\overline{imT^*} \subset (kerT)^{\perp}$

However, I do not know how to prove the equivalence above.

Could you give me some ideas?

 

[Opalg]

Let $$T: X \rightarrow Y$$ be a continuous linear operator between Banach spaces.

Prove that $T$ is surjective $$\iff$$ $$T^*$$ is injective and $$im T^*$$ is closed.

I've proven a "similar" statement, with $$imT^*$$ replaced with $$imT$$.

There I used these facts: $\overline{imT}= ^{\perp}(kerT^*)$ and $\overline{imT^*} \subset (kerT)^{\perp}$

However, I do not know how to prove the equivalence above.

Could you give me some ideas?

A classical theorem of Banach (you'll find it somewhere in his book Opérations linéaires – it's a consequence of the open mapping theorem) says that a bounded linear operator between Banach spaces has closed range if and only if its adjoint also has closed range. I think that will be the key ingredient in your problem.

 

[Linux]

Thank you. Could you tell me where I can find a proof of this theorem, apart from Stefan Banach's book (which is in French)?

 

[Opalg]

(From my phone by Tapatalk)
Try Yosida's Functional Analysis.

 

[Linux]

Thanks. I've just found the book.