# What sequences are in the $l^p$ space?

I am trying to learn about the $l^p$ space for $1\leq p < \infty$. I have read some articles and such but I still don't have any kind of understanding about what sequences are in the $l^p$ space. Or what sequences aren't. Does anyone have a good way to explain this intuitively to me?

Thanks.

The sequences $(x_1,x_2,x_3,\ldots)$ in $l^p$ are those such that the series$$\sum_{n=1}^\infty|x_n|^p$$converges. All other sequences don't belong to $l^p$.

• Does that mean that any sequences that don't absolutely converge would not be in $l^p$? – MathIsHard Aug 24 '17 at 18:39
• @Math4Life No, it does not. For instance, the series $\sum_{n=1}^\infty\frac1n$ doesn't converge absolutely, but the sequence $\left(\frac1n\right)_{n\in\mathbb N}$ belongs to $l^2$. – José Carlos Santos Aug 24 '17 at 18:48
• how is $\frac{1}{n}$ in $l^p$ if it doesn't converge? – MathIsHard Aug 24 '17 at 18:53
• @Math4Life What?! The series $\sum_{n=1}^\infty\left(\frac1n\right)^2=\sum_{n=1}^\infty\frac1{n^2}$ converges! – José Carlos Santos Aug 24 '17 at 18:55
• Ohhh I see. Thank you :) – MathIsHard Aug 24 '17 at 18:56

$\ell^p$ is a sequences space. A sequence in $\ell^p$ is a sequence of p-summable sequences.

More in detail

$$\ell^p:=\{(x_n)_{n \in \mathbb{N}}: \sum\limits_{i=1}^\infty |x_n|^p<+\infty\}$$

This space is a normed space with the norm defined by

$$\|x\|_p:=\left(\sum\limits_{i=1}^\infty |x_n|^p<+\infty \right)^{\frac{1}{p}}$$

Observations:

• Every $(\ell^p,\|f\|_p)$ is a Banach space (complete normed space).
• $\ell^1$ is the space of sequences which have absolutely convergent series.
• Only $\ell^2$ is an Hilbert space (the norm $\|x\|_2$ is induced by an inner product).
• $\ell^\infty$ is the space of bounded sequences

$$\ell^p=\left\{(x_n)_{n\in\mathbb N}\mid \sum_{n\geq 1}|x_n|^p<\infty \right\}.$$

Therefore if $p>1$ then $\left(\sqrt[p]\frac{1}{n}\right)_n\notin\ell^p$ and $\left(\frac{1}{n}\right)_n\in \ell^p$.