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- Jan 26, 2012

- 995

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**Problem**: Suppose that $T\in L(X)$ is a bounded linear operator in a Banach space $X$ such that $\|T\|<1$. Show that $I-T$ is invertible, i.e. has a bounded inverse linear operator and

\[(I-T)^{-1}=\sum_{k=0}^{\infty}T^k.\]

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