Let's first recall that an open mapping $T : X \rightarrow Y$ between two metric spaces is such that for any open $A \subset X$, we have $T(A)$ open in $Y$. The open mapping theorem says the following:
Open mapping theorem: Let $X,Y$ be Banach spaces. If $T : X \rightarrow Y$ is bounded, linear and surjective, then $T$ is open.
The following passage appeared in the book that I am currently studying:
"In our present discussion, the bounded inverse theorem (Open mapping theorem) contributes the following. If $T: X \rightarrow X$ is bounded and linear and $X$ is complete, and if for some $\lambda$, the resolvent $R_\lambda(T)$ exists, and is defined on the whole space $X$, then for that $\lambda$ the resolvent is bounded."
The passage is claiming that $R_\lambda(T) = (T -\lambda I)^{-1}$ is bounded for $\lambda$ satisfying the previous conditions. However, I don't understand how both ideas are related.
Thank you very much in advance!
Edit: The post is answered in the comments.