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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).

1 Answers 1

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For (a), you have $\|\sum_{k = 0}^m F^k - \sum_{k = 0}^n F^k\| \le \sum_{k = m}^n \|F\|^k$

and this last part goes to $0$ as $m,n \to \infty$ because $\sum_{n = 0}^\infty x^n$ converges for $x < 1$ (this is standard calculus). This shows that the partial sums are Cauchy so by the completeness of $L(X)$, the sum converges. For (c),

$\|I - F\|\|(I - F)^{-1}\| \le \|I\|$ so $\|(I - F)^{-1}\| \le \frac{\|I\|}{\|I - F\|}$

but by the triangle inequality you also have

$\|I - F\| \geq \|I\| - \|F\| = 1 - \|F\|$

and the result follows.