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Which of the following sets are compact?

a. The closed unit ball centred at $0$ and of radius $1$ of $\ell_1$ (with the metric $d_1({a_i},{b_i}) = \sum_{i=1}^\infty|a_i - b_i |$).

b. The set of all unitary matrices in $M_2(\Bbb C)$.

c. The set of all matrices in $M_2(\Bbb C)$ with determinant equal to unity.

i know that (b) is true. and (c) is false as it is not bounded. but can't find anything for (a). thank you.

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2 Answers

HINT: In a compact metric space every sequence has a convergent subsequence. For $n\in\Bbb N$ let $e_n$ be the sequence in $\ell_1$ whose $n$-th term is $1$ and whose other terms are all $0$. Does $\langle e_n:n\in\Bbb N\rangle$ have a convergent subsequence?

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For (a) you can try to show the following:

$(\tt{Thm})$ The closed unit ball in a normed space $V$ is compact if and only if $\mathrm{dim}V $ is finite.

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