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I was reading a proof on how any linear combination of compact operators is compact.

Let $U,V: X \to Y$ be compact linear operators and let $\alpha,\beta \in \mathbb{C}$. Then each bounded sequence $(x_n)$ in $X$ contains a subsequence $(x_{n(k)})$ such that $(Ax_{n(k)})$ and $(Bx_{n(k)})$ converge. Then the proof says that due to this $(\alpha A + \beta B)x_{n(k)}$ converges.

But what I dont understand is how we have the same indexing for the subsequence in the case of both operators. I would have thought that as $A$ and $B$ are two operators, the indexing for the subsequences could be different and that we should be considering, say, $(Ax_{n(k)})$ and $(Bx_{n(j)})$ as the convergent subsequences.

So why is this not the case and why does the same indexing apply to both operators when specifying the convergent subsequences?

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Such manipulations are often left implicit in more advanced textbooks. You can choose a subsequence $x_{n_k}$ for the operator $A$ such that $Ax_{n_k}$ converges. Then you can choose a subsequence of $x_{n_k}$ (a bounded sequence, being a subsequence of a bounded sequence) $x_{n_{k_l}}$ such that $Bx_{n_{k_l}}$ converges. Since $Ax_{n_{k_l}}$ is a subsequence of the convergent sequence $Ax_{n_k}$, it also converges so we have a sequence $x_{n_{k_l}}$ such that both $Ax_{n_{k_l}}$ and $Bx_{n_{k_l}}$ converge and this is your required sequence (renamed as $x_{n_k}$).

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  • $\begingroup$ Well, it would be reasonable to write at least "we can find a subsequence that converge", instead of leaving everything implicit... $\endgroup$ Aug 5, 2017 at 21:54
  • $\begingroup$ The quoted text is not wrong nor misleading - indeed each bounded sequence $x_n$ has a subsequence $x_{n_k}$ for which both $Ax_{n_k}$ and $Bx_{n_k}$ converge. It just doesn't specify how you would find such a sequence so in this sense, this is a "very partial proof" whose details are left to the reader... Not that I'm trying to justify such "proofs" - from my point of view, your either don't bother to write a proof for this at all or, if you already do, you should add the extra line to make it less confusing. $\endgroup$
    – levap
    Aug 5, 2017 at 21:59

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