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Suppose $\sum_{j=0}^\infty |x_j|<\infty$ and $h$ a natural number. I have two questions:

  1. How to show that $\sum_{j=0}^\infty x_{j} x_{j+h}< \infty$? My attempt is: since $\sum_{j=0}^\infty |x_j|<\infty$, then $\sum_{j=0}^\infty |x_{j+h}|<\infty$. So $\lim_{j\to \infty}|x_{j+h}|=0$. Consequently, $|x_{j+h}|\leq M< \infty$ , $\forall j$. Thus: $$|\sum_{j=0}^\infty x_j x_{j+h}| \leq \sum_{j=0}^\infty |x_j x_{j+h}| \leq M \sum_{j=0}^{\infty} |x_j| < \infty$$ I think that my attempt is ok. Please, let me know if this ok or wrong. If it is wrong, can you help to show this correclty?
  2. Can we conclude the same ($\sum_{j=0}^\infty x_{j} x_{j+h}< \infty$) if instead of $\sum_{j=0}^\infty |x_j|<\infty$ we assume $\sum_{j=0}^\infty x_j^2<\infty$ ? How to prove?
user346624
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    For the first part what you have done is OK. For the second part just apply Cauchy-Schwarz inequality. – Kavi Rama Murthy Oct 31 '23 at 07:27
  • Are you suggesting that I treat $\sum_{j=0}^{\infty} x_tx_{t+h}$ as the inner product of two infinite vectors? There is a version for finite vectors (https://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality#Rn_-_n-dimensional_Euclidean_space). Does that mean I can generalize this? – user346624 Oct 31 '23 at 07:36
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    The series is absolutely convergent. Apply C-S inequality to the partial sums of $\sum |x_j||x_{j+h}|$ – Kavi Rama Murthy Oct 31 '23 at 07:37
  • Ok. And how about this? https://math.stackexchange.com/questions/1259364/show-that-l2-is-a-hilbert-space I think I can use a CS inequality for the $\ell^2$. I think this is a more direct solution. Am I right? – user346624 Oct 31 '23 at 07:41

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