---
tag: 08P6T2WH6R
kind: example
name: Sequence spaces lp
aliases: [lp-sequence-space]
msc: [46B45]
instance_of: [06GZ956657]
uses: [0BDTTM24DP, 0KNJWEVSMG]
sources:
  - url: https://en.wikipedia.org/wiki/Sequence_space
    locator: ℓp spaces
  - url: https://en.wikipedia.org/wiki/Lp_space
    locator: ℓp spaces and sequence spaces
same_as:
  wikidata: Q289411
  wikipedia: Sequence space
---

## Example

Let $p \ge 1$ be a real number. Let $\ell^p$ be the set of real sequences $x = (x_n)_{n \ge 1}$ with $\sum_{n \ge 1} |x_n|^p < \infty$, and let
$$\|x\|_p = \left( \sum_{n \ge 1} |x_n|^p \right)^{1/p}.$$
Let $\ell^\infty$ be the set of bounded real sequences, with $\|x\|_\infty = \sup_{n \ge 1} |x_n|$.
Then each $\ell^p$, for $1 \le p \le \infty$, is a vector space, $\|\cdot\|_p$ is a norm on it, and $\ell^p$ is a [[Banach space|0BDTTM24DP]].

## Remarks

The norms extend the [[p-norms on R^n|0KNJWEVSMG]] to infinite sequences.
For $1 \le p < q \le \infty$, we have $\ell^p \subseteq \ell^q$, and the inclusion is strict. For example, $(1/n)$ is in $\ell^2$ but not in $\ell^1$.
The space $\ell^2$ has the inner product $\langle x, y \rangle = \sum_n x_n y_n$, and it is a Hilbert space.
The closed unit ball of $\ell^p$ is not compact: the unit vectors $e_k$ satisfy $\|e_j - e_k\|_p = 2^{1/p}$ for $j \neq k$ and $p < \infty$.

## Explanations

### L1 human

The spaces $\ell^p$ are infinite-dimensional versions of $\mathbb{R}^n$ with the p-norm.
A sequence belongs to $\ell^p$ when its terms go to $0$ fast enough for the sum of $|x_n|^p$ to be finite.