Exercise 3, § 1

Let E be a Hilbert space. Denote by $\mathbf{U}(E)$ the group of unitary endomorphisms of E and by $\mathbf{G}\mathbf{L}(E)$ the group of invertible endomorphisms of E.

a) The group $\mathbf{U}(E)$ endowed with the topology induced by the topology of simple convergence on $\mathscr{L}(E)$ is a topological group, denoted by $\mathbf{U}(E)_s$; if E is infinite-dimensional, the group $\mathbf{G}\mathbf{L}(E)$ is not a topological group with the topology induced by the topology of simple convergence.

b) The topology of $\mathbf{U}(E)_s$ coincides with the topology induced by the topology of compact convergence.

c) If E is of countable type, the group $\mathbf{U}(E)_s$ is metrisable and complete. d) Let L be the space of $u\in \mathscr{L}(E)$ such that $u^*=-u$; it is the Lie algebra of the real Lie group $G =\mathbf{U}(E)$ endowed with the topology induced by the Banach-space topology of $\mathscr{L}(E)$ (LIE, III, p. 146, cor. 2). The exponential mapping of L into $\mathbf{U}(E)_s$ is continuous.

e) If E is infinite-dimensional, then $\mathbf{U}(E)_s$ is not a real Lie group of type $\{L\}$. (Consider the subgroup K of $\mathbf{U}(E)_s$ formed by the diagonal unitary endomorphisms in a fixed orthonormal basis of E, and note that K is compact.)

f) The exponential mapping is not a local homeomorphism of L into $\mathbf{U}(E)_s$. (Consider its restriction to the diagonal endomorphisms in a fixed orthonormal basis.)