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A:
1. First go to start menu then type “cmd”, right click on cmd and then “Run As Administrator”.
2. Then type the following lines one after another :
netsh interface tcp set global shutdown=disable
ipconfig /flushdns
3. Restart your computer after typing the above commands.
3.1. You will also have to type the above lines in windows shutdown menu (settings).
4. You should be able to find your sersial number.
Q:
Uniform Norms on a Compact Group
Let $G$ be a compact group. I need to show that if a sequence $\{T_n\}$ of operators on $L^2(G)$ is weakly sequentially compact, then for each $\epsilon > 0$ there exists an $n_0$ such that $\|T_n – T_m\| \leq \epsilon$ for all $n, m \geq n_0$.
I’ve seen the following generalization of a uniform norm on a vector space:
$||T||_U := \sup_{x \in G} \frac{\|Tx\|}{\|x\|}$.
But I don’t know how to adapt it to apply to the above situation.
A:
It is a matter of continuity. By definition, you have $T_n f \to Tf$ for every $f \in L^2$ by the weak compactness. Taking $f$ to be a unit vector in $G$, that’s
$$
\|T_nf-Tf\|^2 = \|T_nf\|^2+\|Tf\|^2-2\textrm{Re}(\langle T_nf,Tf\rangle),
$$
and the right hand side is the distance from the diagonal of the norm square in the matrix of the form
$$
M_n=\int_G\langle f,T_nf\rangle \langle g,T_ng\rangle \;dg.
$$
But now the strong-compactness of $\{M_n\}$ implies the norm-c
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