Surjectivity of $Id-A$ for linear operator $A$ on Banach space with $\|A\|<1$ Let $X$ be Banach space and $A:X\rightarrow X$ linear opeartor such that $\|A\|<1$. It is clear that $Id-A$ is injective. Why is it also surjective?
 A: Hint Prove that $Id+A+A^2+..+A^n+...$ defines an operator which is the inverse of $Id-A$.
A: You know that $B(X)$, the space of bounded linear operators on $X$ is a Banach space. Consider the series
$$
\sum_{n=0}^{\infty} A^n
$$
Since $\|A\| < 1$, it converges absolutely, and thus converges in $B(X)$ to an operator $B$. Now check that $B(I-A) = (I-A)B = I$ and so $(I-A)$ must be surjective.
A: That $I - A$ is both injective and surjective follow from the fact that it is invertible.  Indeed, let $B:X \to X$ be any invertible operator; then we have an operator $B^{-1}$ such that
$BB^{-1} = B^{-1}B = I. \tag{1}$
To see that (1) implies $B$ is injective, suppose that
$Bx_1 = Bx_2 \tag{2}$
for some $x_1, x_2 \in X$.  Then from (1)
$x_1 = Ix_1 = B^{-1}Bx_1 = B^{-1}Bx_2 = Ix_2 = x_2, \tag{3}$
so $B$ is injective.  To see surjectivity, for any $y \in X$ set $x = B^{-1}y$; then again from (1)
$Bx = BB^{-1}y = Iy = y; \tag{4}$
surjectivity proved.
It remains to show $I - A$ is invertible.  But it is well-known that since $\Vert A \Vert < 1$, the series 
$\sum_0^\infty A^n \tag{5}$
converges (it is majorized by $\sum_0^\infty \Vert A \Vert^n$, a convergent real geometric series).  Indeed we have
$(I - A)(\sum_0^\infty A^n) = (\sum_0^\infty A^n)(I - A) = I, \tag{6}$
also well-known.  (6) follows from the algebraic identity
$(I - A)(\sum_0^m A^n) = I - A^{m + 1} \tag{7}$
by letting $m \to \infty$ which forces $A^{m + 1} \to 0$, since $\Vert A^{m + 1} \Vert \le \Vert A \Vert^{m + 1} \to 0$.  This is a very well-known argument which has often been used here on MSE; see my answer to this question for a detailed disussion.  (6) states that the inverse of $I - A$ is $\sum_0^\infty A^n$.  Since $(I - A)^{-1}$ exists, the discussion of the previous paragraph shows that $I - A$ is both injective and also surjective, as sought.
