I need assistance in solving the following:
Let $X$ be a Banach space and $F\in L(X, X)$(space of all bounded linear operators). Suppose $\|F\| \lt 1.$ Let $F^0 = I$.
(a) Using the completeness of $L(X,X)$ show that $\sum_{k=0}^\infty F^k $ converges in $L(X,X)$.
(b) $(I - F)^{-1} = \sum_{k=0}^\infty F^k$.
(c)$\|(I-F)^{-1}\| \leq \frac{\|I\|}{1-\|F\|}$.
I have been able to show (b) and I need assistance with (a) and (c).