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Magazine Problem 4
Show that a metric space \(X\) is compact if and only if it is complete and totally bounded.
Problem 7
Show that a linear operator \(T: X \longrightarrow X\) is compact if and only if for every sequence \(\left(x_{n}\right)\) of vectors of norm not exceeding 1 the sequence \(\left(T x_{n}\right)\) has a convergent subsequence.
Problem 8
Does there exist a surjective compact linear operator \(T: l^{\infty} \longrightarrow I^{*}\) ?
