$\ell^p$Lp2002-02-13 12:08:032005-05-21 10:15:19Definition$\ell^\infty$$\ell^2$% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet $\mathbb{F}$ be either $\mathbb{R}$ or $\mathbb{C}$, and let $p\in\mathbb{R}$ with $p\geq 1$. We define $\ell^p$ to be the set of all sequences $(a_i)_{i\geq 0}$ in $\mathbb{F}$ such that $$\sum_{i=0}^{\infty}|a_i|^p$$ converges.
We also define $\ell^{\infty}$ to be the set of all \PMlinkname{bounded}{BoundedInterval} sequences $(a_i)_{i\geq 0}$ with norm given by $$\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}$$ where $\overline{x}$ denotes the complex conjugate of $x$.
For $p>1$ the (continuous) dual space of $\ell^p$ is $\ell^q$ where $\frac{1}{p} + \frac{1}{q}=1$, and the dual space of $\ell^1$ is $\ell^{\infty}$.
\subsubsection*{Properties}
\begin{enumerate}
\item If $a=(a_0,a_1, \ldots ) \in \ell^p(\mathbb{F})$ for $1\le p< \infty$, then
$\lim_{k\to \infty} a_k =0$.
(\PMlinkname{proof.}{ThenA_kto0IfSum_k1inftyA_kConverges})
\item For $1\le p<\infty$, $\ell^p(\mathbb{F})$ is separable, and $\ell^\infty(\mathbb{F})$
is not separable.
\item Minkowski inequality. If $a,b\in \ell^p(\mathbb{F})$ where $p\ge 1$, then
$$
\Vert a+b \Vert_p \le \Vert a\Vert_p + \Vert b \Vert_p.
$$
\end{enumerate}