Exercise 1, § 1

Let $E = \ell^2_{\mathbf{C}}(\mathbf{N})$ and let $u\in \mathscr{L}(E)$ be the endomorphism such that $u(x_0, . . . , x_n, . . .) = (0, x_0, . . . , x_n, . . .)$ for every $(x_n)_{n\in\mathbf{N}}$ in E. Let $(e_n)$ be the canonical orthonormal basis of E.

a) For every $v\in \mathscr{L}(E)$ such that $\|v\|<1/2$, the vector $e_0$ is not adherent to the image of $u+v$.

b) The open ball in $\mathscr{L}(E)$ of radius $1/2$ centered at $u$ contains no diagonalizable endomorphism of E.