Exercise 15, § 7

The set X of non-empty compact subsets of $\mathbf{C}$ is endowed with the distance defined in TG, IX, p. 91, exerc. 6. Let E be a complex Hilbert space.

a) The mapping $u\mapsto \iota (u)$ from $\mathscr{L}(E)$ into X is continuous. If E is of infinite dimension, it is not continuous when $\mathscr{L}(E)$ is endowed with the topology of simple convergence.

b) The mapping $u\mapsto$ Sp($u$) from $\mathscr{L}(E)$ into X is not continuous.