Is the boundary of the unit sphere in every normed vector space compact? I wanted to ask whether the boundary of the unit sphere in every normed vector space is compact?
I know that this is true for simple examples, but how is it in general?

No. Take
$$
\ell^2 = \{(x_n) \in \mathbb{R} : \sum |x_n|^2 < \infty\}
$$
with the norm
$$
\|(x_n)\| = \sqrt{\sum |x_n|^2}
$$
And consider the vectors
$$
e_n = (0,0,\ldots, 0, 1,0,\ldots )
$$
with $1$ in the $n^{th}$ spot and $0$ elsewhere.
The sequence $\{e_n\}$ is contained in the unit sphere, and has no convergent subsequence.
In general, the unit sphere is compact iff the space is finite dimensional. One way to prove this fact uses the Riesz' Lemma
