Exercise 21, § 4

Let E and F be two Banach spaces. A continuous linear mapping from E into F is said to be a Fredholm operator (or a quasi-isomorphism) if $u^{-1}(0)$ is finite dimensional and $u(E)$ has finite codimension (this implies that $u(E)$ is closed in F and $u$ is a strict morphism); the number $\operatorname{Ind}(u) = \operatorname{codim}(u(E)) - \dim(u^{-1}(0))$ is called the index of $u$. a) Show that $^tu : F' \to E'$ is also a Fredholm operator and that $\operatorname{Ind}(^tu) = - \operatorname{Ind}(u)$. b) If $u : E \to F$ and $v : F \to G$ are two Fredholm operators, then so is $v \circ u : E \to G$ and $\operatorname{Ind}(v \circ u) = \operatorname{Ind}(u) + \operatorname{Ind}(v)$. c) If $w : E \to F$ is a continuous linear mapping with finite rank or with a small enough norm, then $u + w$ is a Fredholm operator and $\operatorname{Ind}(u + w) = \operatorname{Ind}(u)$ (use exerc. 17, c) of IV, p. 65 and 18, b)).