TITLE: The following set is connected or not?
QUESTION [5 upvotes]: Let $E$ be a complex Hilbert space and $S\in \mathcal{L}(E)^+$ (i.e.$S^*=S$ and $\langle Sx\;, \;x\rangle\geq0$ for all $x\in E$).

Let $U=\{x\in E; \langle Sx\;, \;x\rangle=1\}$. Is $U$ a connected set?

Thank you.

REPLY [3 votes]: Let $F = (\ker S)^{\perp}$. Then $S\lvert_F$ is positive definite, and the function $r \colon M \to (0,+\infty)$ given by $r(x) = \sqrt{\langle Sx,x\rangle}$, where $M$ is the unit sphere of $F$, is continuous.
Hence $W = U\cap F = \{ x/r(x) : x \in M\}$ is homeomorphic to $M$, and since unit spheres in complex normed spaces are connected, $W$ is connected [with the caveat that some people consider $\varnothing$ disconnected, for them we would need to exclude $S = 0$]. And since
$$\langle S(x+y),x+y\rangle = \langle Sx,x\rangle$$
for $y \in \ker F$, it follows that $U$ is homeomorphic to $(\ker S) \times W$, hence connected.