$T$ is compact iff $\langle Te_i,e_j\rangle\to 0$ as $i,j\to\infty$
71011
Criteria of compactness of an operator
140756
Image of closed unit ball under a compact operator
48498
Fourier coefficients $\langle x,e_k\rangle$ are at countable
19741
If $(\mu_t)_{t\in I}$ and $(\mu_t\ast\nu_t)_{t\in I}$ are tight, then $(\nu_t)_{t\in I}$ is tight
127173
Closed subset of compact set is compact
8698
Show the Volterra Operator is compact using only the definition of compact
100257
Converse to Hilbert basis theorem
2056
A space is $T_{0}$ if and only if the closures of singletons are distinct
31124
The closure of $c_{00}$ is $c_{0}$ in $\ell^\infty$
39076
Contraction of compact sets
43376
If $A$ is compact and $B$ is closed, show $d(A,B)$ is achieved
146455
Compactness of the set of all unitary matrices in $M_2(\mathbb{C})$
72209
show that torus is compact
80735
Show that $C_0([a, b], \mathbb{R})$ is not $\sigma$-compact
86481
Let $H$ be an infinite-dimensional separable Hilbert space, show that a Hilbert-Schmidt operator is compact
119352
compact and self adjoint square root of an operator
31919
What can we say about the inner product $\langle f(t),f'(t) \rangle$?