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Thanks. I've just found the book.

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)?

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}=...

Let $$(X, d)$$ be a metric space, $$AE_0(X) = \{ u : X \rightarrow \mathbb{R} \ : \ u^{-1} (\mathbb{R} \setminus \{0 \} ) \ \ \text{is finite}, \ \sum_{x \in X} u(x)=0 \}$$, for $$x,y \in X, \ x \neq y, \ m_{xy} \in AE_0(X), \ \ m_{xy} (x)=1, \ m_{xy}(y)=-1, \ m_{xy}(z)=0$$ for $$z \neq x...