It's well known that on a Banach space, the spectrum of each bounded operator is compact in $\mathbb C$. What about a general normed vector space? Is there a counterexample if we don't assume completeness?
(The initial motivation: I was looking at the following exercise from Kreyszig: Let $T: X \to X$ be a compact operator on a normed space $X$. If $\dim X = \infty$, then $0 \in \sigma(T)$. For a Banach space, we can use the closedness of the spectrum to get 0 as a limit point of the spectrum. But I don't see how this can be done in general)