User:Caliburn/s/fa/2

Theorem
Let $\struct {X, \norm \cdot_X}$ be a Banach space.

Let $\norm \cdot$ be the norm of a bounded linear transformation.

Let $T : X \to X$ be a bounded linear operator with $\norm T < 1$.

Then:


 * $(1) \quad$ $I - T$ is invertible with inverse $\paren {I - T}^{-1}$
 * $(2) \quad$ $\paren {I - T}^{-1}$ is bounded.
 * $(3) \quad$ $\ds \paren {I - T}^{-1} = \sum_{k \mathop = 0}^\infty T^k$
 * $(4) \quad$ $\norm {\paren {I - T}^{-1} } \le \paren {1 - \norm T}^{-1}$