Exercise 4, § 3

Soit E an infinite-dimensional complex hermitian space.

a) The group $\mathbf{U}(E)$ of unitary operators on E, endowed with the topology induced by the norm of $\mathscr{L}$ (E), is arcwise connected. (Prove that if $u$ is unitary, there exists a hermitian operator $v$ such that $u=$ exp($iv$)$.$) b) The group of linear isomorphisms of E is arcwise connected. (Use the polar decomposition, cf. I, p. 139, déf. 3.)

c) For every $n\in \mathbf{Z}$, the set of Fredholm endomorphisms of E of index $n$ is open and connected. It is not empty.