| Version 17 |
Version 16 |
| 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. |
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. |
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| We also define $\ell^{\infty}$ to be the set of all \PMlinkname{bounded}{BoundedInterval} sequences $(a_i)_{i\geq 0}$ with norm given by $$\Vert (a_i)\Vert_{\infty} = \operatorname{sup}\{ |a_i|:i\geq 0\}.$$ |
We also define $\ell^{\infty}$ to be the set of all \PMlinkname{bounded}{BoundedInterval} sequences $(a_i)_{i\geq 0}$ with norm given by $$\Vert (a_i)\Vert_{\infty} = \operatorname{sup}\{ |a_i|:i\geq 0\}.$$ |
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| By defining addition and scalar multiplication pointwise, $\ell^p(\mathbb{F})$ and |
By defining addition and scalar multiplication pointwise, $\ell^p(\mathbb{F})$ and |
| $\ell^\infty(\mathbb{F})$ have a natural vector space stucture. |
$\ell^\infty(\mathbb{F})$ have a natural vector space stucture. |
| That the sum of two elements on $\ell^p(\mathbb{F})$ is again an element |
That the sum of two elements on $\ell^p(\mathbb{F})$ is again an element |
| in $\ell^p(\mathbb{F})$ follows from Minkowski inequality |
in $\ell^p(\mathbb{F})$ follows from Minkowski inequality |
| (see below). |
(see below). |
| We can make $\ell^p$ into a normed vector space, by defining the norm as $$\Vert (a_i)\Vert_p = (\sum_{i=0}^{\infty}|a_i|^p)^{1/p}.$$ |
We can make $\ell^p$ into a normed vector space, by defining the norm as $$\Vert (a_i)\Vert_p = (\sum_{i=0}^{\infty}|a_i|^p)^{1/p}.$$ |
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| 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 $x$. |
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 $x$. |
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| 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 $\ell^{\infty}$. |
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 $\ell^{\infty}$. |
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| \subsubsection*{Properties} |
\subsubsection*{Properties} |
| \begin{enumerate} |
\begin{enumerate} |
| \item If $a=(a_0,a_1, \ldots ) \in \ell^p(\mathbb{F})$ for $1\le p< \infty$, then |
\item If $a=(a_0,a_1, \ldots ) \in \ell^p(\mathbb{F})$ for $1\le p< \infty$, then |
| $\lim_{k\to \infty} a_k =0$. |
$\lim_{k\to \infty} a_k =0$. |
| (\PMlinkname{proof.}{ThenA_kto0IfSum_k1inftyA_kConverges}) |
(\PMlinkname{proof.}{ThenA_kto0IfSum_k1inftyA_kConverges}) |
| \item For $1\le p<\infty$, $\ell^p(\mathbb{F})$ is separable, and $\ell^\infty(\mathbb{F})$ |
\item For $1\le p<\infty$, $\ell^p(\mathbb{F})$ is separable, and $\ell^\infty(\mathbb{F})$ |
| is not separable. |
is not separable. |
| \item Minkowski inequality. If $a,b\in \ell^p(\mathbb{F})$ where $p\ge 1$, then |
\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. |
\Vert a+b \Vert_p \le \Vert a\Vert_p + \Vert b \Vert_p. |
| $$ |
$$ |
| \item The assumption $p\ge 1$ is motivated. If $0<p<1$, $\ell^p(\mathbb{F})$ (defined |
\item The assumption $p\ge 1$ is motivated. If $0<p<1$, $\ell^p(\mathbb{F})$ (defined |
| as above) is not even a metric space \cite{friedman}. |
as above) is not even a metric space \cite{friedman}. |
| \end{enumerate} |
\end{enumerate} |
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{friedman} |
\bibitem{friedman} |
| A. Friedman, |
A. Friedman, |
| \emph{Foundations of Modern Analysis}, |
\emph{Foundations of Modern Analysis}, |
| Dover publications, 1982. |
Dover publications, 1982. |
| \end{thebibliography} |
\end{thebibliography} |
