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The Class of All Square-Summable Real Sequences... l^2(N) ... Sohrab Example 2.3.52 ... ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,891
I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).

I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...

I need help with an aspect of Example 2.3.52 ...

The start of Example 2.3.52 reads as follows ... ...



Sohrab - Start of Example 2.3.52 ... .png






In the above Example from Sohrab we read the following:

" ... ... Then, given any sequences \(\displaystyle x = (x_n), \ y = (y_n) \in l^2 ( \mathbb{N} )\), the series \(\displaystyle \sum_{ n = 1 }^{ \infty } x_n y_n \) is absolutely convergent ... ..."



My question is as follows:

How/why, exactly, given any sequences \(\displaystyle x = (x_n), \ y = (y_n) \in l^2 ( \mathbb{N} )\) ...

... does it follow that the series \(\displaystyle \sum_{ n = 1 }^{ \infty } x_n y_n \) is absolutely convergent ... ...?




Help will be much appreciated ... ...

Peter
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,681
In the above Example from Sohrab we read the following:

" ... ... Then, given any sequences \(\displaystyle x = (x_n), \ y = (y_n) \in l^2 ( \mathbb{N} )\), the series \(\displaystyle \sum_{ n = 1 }^{ \infty } x_n y_n \) is absolutely convergent ... ..."



My question is as follows:

How/why, exactly, given any sequences \(\displaystyle x = (x_n), \ y = (y_n) \in l^2 ( \mathbb{N} )\) ...

... does it follow that the series \(\displaystyle \sum_{ n = 1 }^{ \infty } x_n y_n \) is absolutely convergent ... ...?
This comes from the inequality $(\dagger)$, where it is proved that \(\displaystyle \sum_{ n = 1 }^{ \infty }| x_n y_n| \leqslant \|x\|_2\|y\|_2 < \infty. \)
 

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,891
This comes from the inequality $(\dagger)$, where it is proved that \(\displaystyle \sum_{ n = 1 }^{ \infty }| x_n y_n| \leqslant \|x\|_2\|y\|_2 < \infty. \)


Appreciate the help, Opalg ...

Thanks ...

Peter