!
I am still confused about the range of $\sigma(I_{\omega})$.Since $\|x_n\|_2\to 0,$we have $\|x_n\|\to 0$,$\sigma(I_{\omega})=0,$then $J=0$,it is trivial.
Is my understanding correct?
2026-03-26 10:07:07.1774519627
image of some ideal under the quotient map
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No, it is not true that if $\|x_n\|_2\to0$, then $\|x_n\|\to0$.
For instance, suppose that $k(n)=n$, $x_n=E_{11}$ for all $n$. Then $\|x_n\|_2=1/\sqrt n$, while $\|x_n\|=1$.