Let $x=(x_1, x_2, \cdots) \in \ell^{\infty} = (\ell^1)^*$.
Let $s_n = \sum_{k=1}^n x_k e_k$ be the $n$-th partial sum.
(1) Show that $s_n \stackrel{\ast}{\rightharpoonup}x$ (i.e. show the the weak $^*$ convergence).
(2) Does the net $s_F= \sum_{k\in F} x_k e_k$ of finite partial sums converge int the weak$^*$ topology to $x$?
Things I know:
For (1):
We have that $\ell^1 = c_0^*$.
Since $(e_n) \subset \ell^1$, $(e_n)$ converges to $0$ in the weak $^*$ topology.
Since $s_n$ is finite, we know that each $s_n \in \ell^1.$.
So we want to show that $\lim_{n\rightarrow \infty} s_n= x$.
But this above seems obvious. I'm sure I missed something important here, since I haven't really used the the definitions of $\ell^1$ and $\ell^{\infty}$…
So please let me know how to proceed from here.
For (2),
I'm not sure about this one. My guess is yes. But I don't have any clues on how to prove that.
Any help is appreciated!
Thank you!
Best Answer
Let $(y_k) \in \ell^{1}$. Then $s_n((y_k))= \sum\limits_{k=1}^{n}x_ky_k \to \sum\limits_{k=1}^{\infty}x_ky_k$, Note that the series $\sum\limits_{k=1}^{\infty}x_ky_k$ converges absolutely. This proves the first part. The second part follows from the fact that for any $\epsilon >0$ we can find $N$ with $\sum\limits_{k=N}^{\infty}|x_k||y_k| <\epsilon$.