Let $X,Y$ be Banach spaces. A bounded linear operator $T:X \to Y$ is a Fredholm operator if $\text{KerT}$ and $Y/T(X)$ are finite dimensional. We define the index of $T$ as the difference of dimensions: $$ \operatorname{ind}T = \text{dim}(\ker T)-\dim \left(Y/T(X)\right). $$
Consider $T = S + W$, where $S$ is an isomorphism and $W$ is compact. Then $T$ is a Fredholm operator and the index of $T$ is zero. Why?