Exercise 2, § 6

Let $H$ be a separated prehilbertian space of infinite dimension. Let $(e_i)_{i\geqslant 0}$ be an orthonormal family of vectors of $H$ which generates a dense subspace of $H$. For every integer $n\geqslant 1$, denote by $V_n(H)$ the subspace of $H^n$ consisting of the sequences $(u_1, . . . , u_n)$ which are orthonormal.

a) Prove that there exists one and only one continuous linear mapping $T: H\rightarrow H$, such that $T(e_i) =e_{i+1}$ for every integer $i$. Prove that one has $\|T(x)\|=\|x\|$ for every $x\in H$.

b) Let $H'$ be the subspace of H formed by the vectors orthogonal to $e_i$, for $i < n$. Prove that the following conditions are equivalent: (i) $x$ and $T^n(x)$ are linearly dependent ; (ii) $T(x) =x$; (iii) $x\in H'$.

c) Prove that there exists a unique homotopy $\sigma : V_n(H)\times \mathbf{I}\rightarrow V_n(H)$ such that, for every $(u_1, . . . , u_n)\in V_n(H)$ and every $t\in \mathbf{I},\sigma ((u_1, . . . , u_n), t)$ is the sequence deduced from the family $(1-t)(u_1, . . . , u_n) +t(T^n(u_1), . . . ,T^n(u_n))$ by the orthonormalization process (EVT, V, p. 23, prop. 6).

d) Prove that there exists a unique homotopy $\tau : V_n(H')\times \mathbf{I}\rightarrow V_n(H)$ such that, for every $(u_1, . . . , u_n)\in V_n(H')$ and every $t\in \mathbf{I},\sigma ((u_1, . . . , u_n), t)$ is the sequence deduced from the family $(1-t)(u_1, . . . , u_n) +t(e_1, . . . , e_n)$ by the orthonormalization process.

e) Prove that the space $V_n(H)$ is contractible.