Let $F(A)=\{x^*Ax\mid x\in \mathbb{C}^n,\|x\|=1\}$ where $A\in M_n(\mathbb{C}).$ Describe $F(A)$ when $A$ is Hermitian.
Solution 1:
You need compactness and connectedness: the sphere $S=\{ x\in \mathbb C^n : \|x\| = 1\}$ is connected and compact, while $$f:S\to \mathbb R, \ \ x\mapsto x^* Ax$$
is continuous, so the image is connected and compact.