Exercise 13, § 2

Let E be an infinite dimensional hilbertian space satisfying the first axiom of countability and $(e_n)_{n \in \mathbf{Z}}$ an orthonormal basis of E whose set of indices is the set of rational integers. Let u denote the isometry from E onto itself such that $u(e_n) = e_{n+1}$ for all $n \in \mathbf{Z}$, and put

a) Let B be the unit ball and S the unit sphere in E. Show that the restriction of f to B is a homeomorphism from B onto itself (observe that the restriction of u to S is a homeomorphism from S onto itself), and that there does not exist any point $x_0 \in B$ such that $f(x_0) = x_0$ (express $x_0$ in terms of its coordinates with respect to $(e_n)$). b) For every $x \in B$, let $g(x)$ be the point of intersection of S with the half-line with origin $f(x)$ passing through x. Show that g is a continuous mapping from B onto S, such that $g(x) = x$ for all $x \in S$ (compare with GT, VI, § 2, exerc. 8). Deduce that there exists $x_0 \in S$ and a continuous mapping h from $S \times [0, 1]$ onto S such that $h(x, 0) = x_0$ and $h(x, 1) = x$ for all $x \in S$.