Let $\{e_n\}$ be an orthonormal basis for $l^2$ and $\{\alpha_n\} \in \ell^{\infty}$. Define $Ae_n=\alpha_n e_n$ on $l^2$. Find the condition for $A$ to be compact. My work: I want to show that in order for $A$ to be compact, we must have $\alpha_n \rightarrow 0$. Suppose the contrary. Then there exists some $\epsilon &gt;0 $ such that we can find some sequence $(\alpha_{n_k})_k$ such that $(\alpha_{n_k})_k&gt; \epsilon$. I want to show that in this case, we can't find a convergent subsequence for $(Ae_{n_k})=(\alpha_{n_k}e_n)_k$. And I got stuck here. Not sure how to show that there doesn't exist a convergent subsequence. I feel like I'm missing some facts in real analysis. Any help is appreciated! Thank you.
Let $p\in[1,\infty]$, $a\in\ell^\infty$ and $$T:\ell^p\to\ell^p\;,\;\;\;x\mapsto ax.$$ Note that $\left\|T\right\|=\left\|a\right\|_{\ell^\infty}$. It's easy to see that if $a\in\ell^1$, then $T$ is nuclear and hence compact.  Can we more generally conclude that $T$ is compact?  Assume $p&lt;\infty$. The intuitive approach would be to define $$T_n:\ell^p\to\ell^p\;,\;\;\;x\mapsto(a_1x_1,\ldots,a_nx_n,0,\ldots)$$ for $n\in\mathbb N$ and to show that $\left\|T_n-T\right\|\to0$. Now, $$\left\|(T_n-T)x\right\|^p=\sum_{i=n+1}^\infty|a_ix_i|^p\tag1$$ for all $x\in\ell^p$ and $n\in\mathbb N$. The right-hand side of $(1)$ clearly converges to $0$ as $n\to\infty$, but this only yields convergence in the strong operator topology. On the other hand, if $(a_n)_{n\in\mathbb N}$ would be nonincreasing and $a_n\to0$ (e.g. $a_n=\frac1n$), then $$\left\|(T_n-T)x\right\|^p\le|a_{n+1}|^p\left\|x\right\|_{\ell^p}^p\tag2$$ for all $x\in\ell^p$ and $n\in\mathbb N$ and we could immediately conclude $\left\|T_n-T\right\|\to0$. EDIT: Maybe the general result can be obtained from the convergence in strong operator topology (which holds by $(1)$) by noting that $\ell^p$ has the approximation property; but I'm not sure whether this is true.
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