- #1

JD_PM

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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.

Thanks.