Suppose $T$ is a bounded linear operator on a banach space $X$, such that $\|I-T\| < 1$. Show that $T$ has an inverse and it is bounded.
To show that an inverse exists, we have to show that $T$ is injectice and surjective. Now, I can prove tha injective part, but cannot prove the surjectivity, and how is the inverse bounded.
I know a similar question was asked here but the answer there did not help at all. I also cannot ask for an explanation in the comments either because my reputation is not yet 50.
Edit: $I$ is the identity operator.