Let $(a_n)\subset\mathbb{C}$ be bounded. Then $T_a\colon\ell^p\to\ell^p, \ (T_a x)_n:=a_n x_n$ compact iff $a_n\to0$.
124239
example of compact operator
142901
"The closure of the unit ball of $C^1[0, 1]$ in $C[0, 1]$" and its compactness
12467
Why does this proof work: Closed unit ball in $C_0$ is not compact
80735
Show that $C_0([a, b], \mathbb{R})$ is not $\sigma$-compact
23231
Convergence of $A_nT$ to $AT$ in operator norm for compact $T$
47810
$C(X)$ is separable when $X$ is compact?
83337
If $A ∈M_n(C)$ is such that $(Ax,x) = 0$ for all $x ∈C^n$, then $A = 0$
113151
Does $\sum\limits_{i=1}^{\infty}|a_i||x_i| < \infty$ whenever $\sum\limits_{i=1}^{\infty} |x_i| < \infty $ imply $(a_i)$ is bounded?
56628
If $f: D(0,1)\to \mathbb C$ is holomorphic, then there is a sequence $\{z_n\}$ in $D(0,1)$ such that $|z_n| \to 1$ and $\{f(z_n)\}$ is bounded
102478
$X_n\leq Y_n$ implies $\liminf X_n \leq \liminf Y_n$ and $\limsup X_n \leq \limsup Y_n$
28391
Prove the compactness of a set B
74501
$\delta : C([0,1])\to\mathbb{R}$, $f\mapsto f(0)$, with sup-norm is bounded but unbounded with one-norm.
29114
If $f \in L^{p_1}(E) $ is bounded then $f\in L^{p_2}$ for any $p_2>p_1$
128558
Inequality between $\ell^p$-norms
83311
Compactness of $A:=${$f \in C[0,1], |f|_\infty \le K, |f'|_\infty \le M$}
18596
A set $A \subset l_1$ is compact if and only if closed, bounded, and one other condition
148994
$K \subset \mathbb{R}^n$ is compact iff it is closed and bounded