Let $p(.)$ be an equivalent norm to the usual norm on $\ell_1$ such that
$$\limsup\limits_{n\to\infty} p(x_n+x)=\limsup\limits_{n\to\infty}p(x_n)+p(x)$$ for every $w^*-$null sequence $(x_n)$ and for all $x\in\ell_1,$ moreover, let $$\rho_{k}(x)=p(x)+\lambda\gamma_{k}\sum\limits_{n=k}^{\infty}|x_n|,$$ where, $(\gamma_{k})$ be any non-decreasing sequence in $(0,1)$ and $\lambda >0$. I'd like to prove for every $w^*-$null sequence $(x_n)$ and for all $x\in\ell_1,$
$\limsup\limits_{n\to\infty}\rho_k(x_n+x)=\limsup\limits_{n\to\infty}\rho_k(x_n)+\rho(x)$ .
**My attempt is the following**
\begin{align}
\limsup\limits_{n\to\infty}\rho_k(x_n+x)
=\limsup\limits_{n\to\infty} p(x_n+x) +\limsup\limits_{n\to\infty}\lambda\gamma_{k}\sum\limits_{n=k}^{\infty}|x_n+x| \\
=\limsup\limits_{n\to\infty}p(x_n)+p(x) +\limsup\limits_{n\to\infty}\lambda\gamma_{k}\sum\limits_{n=k}^{\infty}|x_n+x|\\
\end{align}
Now I could not proceed to prove, any ideas or hints would be greatly appreciated.
Thanks in advance
First of all, I suppose that you consider $\ell_1$ as dual to $c_0$, so by $w^*-$null sequence in $\ell_1$ you mean such a sequence $x_n$ that is norm-bounded and goes to zero coordinate-wise. In your question you are using letter $x_n$ in two different senses: as elements of $\ell_1$ and as coordinates of an element $x$. This leads to a confusion. If you denote coordinates of $x$ as $x_n$ and consider vectors $y_n = (y_{n,1}, y_{n,2}, \ldots)$, then the formula you are going to demonstrate is $\limsup\limits_{n\to\infty}\rho_k(y_n+x)=\limsup\limits_{n\to\infty}\rho_k(y_n)+\rho(x)$
The hint to this exercise is: if a sequence of vectors $y_n \in \ell_1$ is norm-bounded, goes to zero coordinate-wise and if there exists the limit $\lim_{n \to \infty} \|y_n\|$, then there exists $\lim_{n \to \infty} \|x +y_n\|$, and
$$\lim_{n \to \infty} \|x +y_n\| = \lim_{n \to \infty} \sum_{j=1}^\infty |x_j + y_{n,j}| = \|x\| + \lim_{n \to \infty} \|y_n\|.$$
Dear Prof Vladimir Kadets ,
I am really appreciate your attention,thank you very much for your respond, it is very excellent, yes I consider $\ell_1$ as a dual of $c_0$ but unfortunately I a little confused abut the proof of
$$\lim_{n \to \infty} \|x +y_n\| = \lim_{n \to \infty} \sum_{j=1}^\infty |x_j + y_{n,j}| = \|x\| + \lim_{n \to \infty} \|y_n\|.$$
please let me know how proceed to prove this statement
yours Sincerely
At first, for an $\epsilon > 0$ find such a $k$, that $\sum_{j=k+1}^\infty |x_j| N$ $$\sum_{j=1}^k |y_{n,j}| N$, using $|\|x\| - \sum_{j=1}^k |x_j||
Dear Anna Vishnyakova ,
I am really appreciate your respond, It is really useful
Best wishes
Do you people can actually understand all that mumbo jumbo with LaTeX or MathJaX written here? I can't....Is there something that can be done so that this site will translate all that into normal, usual symbols in mathematics?
I agree, it would be convenient, if this site had a visualization of LaTeX formulas.
On the other hand, a professional mathematician writes in LaTeX very often, and thus usually is able to understand LaTeX without compilation. In many instances such understanding is very useful, for example for mathematical communication in chat. Also, I should remark, that to learn elements of LaTeX is not a big deal.
Oh, I am very proficient both in LaTeX and MathJaX and I use them very often. Yet, with all those /, $ and etc. symbols it is, at least for me, very difficult to understand what is going on. A pity this site, unlike Stackexchange mathematics, can't handle LaTex-like symbols...
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