Sequence spaces lp

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.

Remarks

The norms extend the p-norms on R^n 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.