Let $\mathbb{F}$ be either $\mathbb{R}$ or $\mathbb{C}$ and let $p\in\mathbb{R}$ with $p\geq 1$ We define $\ell^p$ to be the set of all sequences $(a_i)_{i\geq 0}$ in $\mathbb{F}$ such that $$\sum_{i=0}^{\infty}|a_i|^p$$ converges.

We also define $\ell^{\infty}$ to be the set of all bounded sequences $(a_i)_{i\geq 0}$ with norm given by $$\Vert (a_i)\Vert_{\infty} = \operatorname{sup}\{ |a_i|:i\geq 0\}.$$

By defining addition and scalar multiplication pointwise, $\ell^p(\mathbb{F})$ and $\ell^\infty(\mathbb{F})$ have a natural vector space stucture. That the sum of two elements on $\ell^p(\mathbb{F})$ is again an element in $\ell^p(\mathbb{F})$ follows from Minkowski inequality (see below). We can make $\ell^p$ into a normed vector space, by defining the norm as $$\Vert (a_i)\Vert_p = (\sum_{i=0}^{\infty}|a_i|^p)^{1/p}.$$

The normed vector spaces $\ell^{\infty}$ and $\ell^p$ for $p\geq 1$ are complete under these norms, making them into Banach spaces. Moreover, $\ell^2$ is a Hilbert space under the inner product $$\langle (a_i),(b_i)\rangle = \sum_{i=0}^{\infty}a_i \overline{b_i}$$ where $\overline{x}$ denotes the complex conjugate of $x$

For $p>1$ the (continuous) dual space of $\ell^p$ is $\ell^q$ where $\frac{1}{p} + \frac{1}{q}=1$ and the dual space of $\ell^1$ is $\ell^{\infty}$

Properties

1. If $a=(a_0,a_1, \ldots ) \in \ell^p(\mathbb{F})$ for $1\le p< \infty$ then $\lim_{k\to \infty} a_k =0$ (proof.)
2. For $1\le p<\infty$ $\ell^p(\mathbb{F})$ is separable, and $\ell^\infty(\mathbb{F})$ is not separable.
3. Minkowski inequality. If $a,b\in \ell^p(\mathbb{F})$ where $p\ge 1$ then $$ \Vert a+b \Vert_p \le \Vert a\Vert_p + \Vert b \Vert_p. $$