Understanding why ##(y_n)_n## is a bounded sequence

[JD_PM]

Suppose $(y_n)_n$ is a sequence in $\mathbb{C}$ with the following property: for each sequence $(x_n)_n$ in $\mathbb{C}$ for which the series $\sum_n x_n$ converges absolutely, also the series $\sum_n \left(x_ny_n\right)$ converges absolutely. Can you then conclude that $(y_n)_n$ is a bounded sequence?

Well, I am trying to understand two of the answers I got on Math Stack Exchange for this (see for more details: https://math.stackexchange.com/questions/3103987/concluding-whether-y-n-n-is-a-bounded-sequence ).

- In one of the answers, David C. Ullrich states:

Suppose $y_n$ is unbounded. There is a subsequence $y_{n_k}$ with $|y_{n_k}|>k^3$. Define $x_n$ by saying

$$x_{n_k}=1/k^2,$$

$x_n=0$ if $n\ne n_k$. Then $\sum x_n$ converges absolutely, but $\sum y_n x_n$ diverges, since the terms do not even tend to $0$. (Because $|y_{n_k}x_{n_k}|>k$).

Here I do not see why $y_n$ being unbounded implies that there is a subsequence $y_{n_k}$ with $|y_{n_k}|>k^3$

- In another answer, Rigel states:

You can prove this result by using the Banach-Steinhaus theorem.

More precisely, let $A_n\colon \ell^1\to\mathbb{C}$ be the functional defined by
$$
A_n x := \sum_{j=1}^n x_j y_j.
$$
As is customary, $\ell^1$ denotes the set of complex sequences $x = (x_1, x_2, \ldots)$ such that $\|x\|_1 := \sum_{j=1}^\infty |x_j| < +\infty$.

Clearly $|A_n x| \leq C_n \|x\|_1$, where $C_n := \max\{|y_1|, \ldots, |y_n|\}$.
Hence, $A_n \in (\ell^1)^* = \ell^\infty$ and it is not difficult to check that $\|A_n\|_* = C_n$.

By assumption, for every $x\in\ell^1$ there exists the limit
$$
Ax := \lim_n A_n x = \sum_{j=1}^\infty x_j y_j.
$$
Then, by the Banach-Steinhaus theorem, $A\in (\ell^1)^*$ and
$$
\|A\|_* \leq \liminf_n \|A_n\|_{*} = \sup_{j\in\mathbb{N}} |y_j| < \infty,
$$
so that $(y_j)$ is bounded.

I have been reading about Banach–Steinhaus theorem (https://en.wikipedia.org/wiki/Uniform_boundedness_principle) but still do not see how the theorem is used to see if $(y_j)$ is bounded. Please either explain the general idea or recommend a book where I could read about it. I am currently using Rudin's and Abbott's (note I am a beginner).

Thanks.

 

[mfb]

The sequence can't be bound by 1, there has to be an element such that $|y_k|>1$. Take the first one as first element of your subsequence. The sequence can't be bound by 2³, there has to be an element such that $|y_l|>2^3$, even if you look at l>k only . Take that as second element of your subsequence. The sequence can't be bound by 3³, there has to be an element such that $|y_m|>3^3$, even if you look at m>l only . Take that as third element of your subsequence. And so on.

 

[JD_PM]

The sequence can't be bound by 1, there has to be an element such that $|y_k|>1$. Take the first one as first element of your subsequence. The sequence can't be bound by 2³, there has to be an element such that $|y_l|>2^3$, even if you look at l>k only . Take that as second element of your subsequence. The sequence can't be bound by 3³, there has to be an element such that $|y_m|>3^3$, even if you look at m>l only . Take that as third element of your subsequence. And so on.

Sorry but I still do not understand the whole process. I get that if we suppose $y_n$ is unbounded then there has to be an element such that $|y_k|>1$. But once you start with 'The sequence can't be bound by 2³' I get lost. I mean, why the 2³ factor?

 

[mfb]

It is an arbitrary choice. I picked it because the answer you quoted uses k³ as comparison: 1³, 2³, 3³. There are many more sequences that work.