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\begin{document}

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\section*{sequence space}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{sequence_spaces}{}\section*{{Sequence spaces}}\label{sequence_spaces}

\noindent\hyperlink{sequence_spaces_2}{Sequence spaces}\dotfill \pageref*{sequence_spaces_2} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{Generalisations}{Generalisations}\dotfill \pageref*{Generalisations} \linebreak
\noindent\hyperlink{in_constructive_mathematics}{In constructive mathematics}\dotfill \pageref*{in_constructive_mathematics} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{sequence_spaces_2}{}\subsection*{{Sequence spaces}}\label{sequence_spaces_2}

A classical \emph{sequence space} is a [[vector space]] of [[sequences]] of [[real numbers]], equipped with a [[p-norm]] that makes it a [[normed vector space]]. More generally one may consider spaces of functions on any set.

Specific sequence spaces are usually known through their symbolic names, such as `$c_0$' and `$l^p$', that appear below. The term `sequence space' is useful as a general name without symbols in it.

The sequences spaces are basic examples of [[topological vector spaces]]. They all have a discrete flavour that (maybe) makes them easy to understand, but they are not actually [[discrete spaces]].

The generalization of sequences space to spaces of functions on more general [[measure spaces]] are the \emph{[[Lebesgue spaces]]}.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

Fix a [[set]] $N$; typically, $N$ is the set $\mathbb{N}$ of [[natural numbers]], but this is not necessary for the basic concepts. Sometimes one uses the set $\mathbb{Z}$ of [[integers]] (which is the [[underlying set]] of an [[abelian group]], useful for some purposes), which of course is [[bijective]] with $\mathbb{N}$. For the simplest examples, let $N$ be a [[finite set]].

Also fix a [[topological vector space]] $K$; typically, $K$ is either the space $\mathbb{C}$ of [[complex numbers]] or the space $\mathbb{R}$ of [[real numbers]]. We will assume below that $K$ is at least a [[Banach space]]; but since much of the point of the sequence spaces is to be simple examples of Banach spaces, you probably want something familiar as $K$.

We will think of a [[function]] from $N$ to $K$ as a \textbf{$K$-valued $N$-sequence}, or simply a \textbf{sequence}. The various sequence spaces will be [[subsets]] of the [[function set]] $K^N$ of all sequences. In general, if `$X$' is the symbol for a sequence space, then we may specify $N$ and $K$ by writing `$X(N,K)$' (or a variation thereon), but often this is suppressed.

\begin{defn}
\label{}\hypertarget{}{}
$l^1$ is the space of \textbf{absolutely summable sequences}:

\begin{displaymath}
\sum_k {|a_k|} \lt \infty .
\end{displaymath}
We equip $l^1$ with the \textbf{$l^1$-norm}

\begin{displaymath}
{\|a\|_1} \coloneqq \sum_k {|a_k|} .
\end{displaymath}
\end{defn}
This is a [[Banach space]].

\begin{defn}
\label{}\hypertarget{}{}
$l^2$ is the space of \textbf{absolutely square-summable sequences} (or, over a [[real field]], simply \emph{square-summable sequences}):

\begin{displaymath}
\sum_k {|a_k|^2} \lt \infty .
\end{displaymath}
We equip $l^2$ with the \textbf{$l^2$-norm}

\begin{displaymath}
{\|a\|_2} \coloneqq \sqrt{\sum_k {|a_k|^2}} .
\end{displaymath}
\end{defn}
This is also a Banach space; in fact, it's a [[Hilbert space]] (assuming that $K$ is). Furthermore, \emph{every} Hilbert space (over $K$ a [[field]]) arises in this way, up to [[isometric isomorphism]], using an [[orthonormal basis]] for $N$.

More generally, for $0 \lt p \lt \infty$:

\begin{defn}
\label{}\hypertarget{}{}
$l^p$ is the space of \textbf{absolutely $p$th-power--summable sequences}:

\begin{displaymath}
\sum_k {|a_k|^p} \lt \infty .
\end{displaymath}
We equip $l^p$ with the \textbf{$l^p$-norm}

\begin{displaymath}
{\|a\|_p} \coloneqq \sqrt[p]{\sum_k {|a_k|^p}} .
\end{displaymath}
\end{defn}
This is at least an $F$-[[F-space|space]], which is a Banach space iff $p \geq 1$. (For $p \lt 1$, the `norm' is not really a norm in the sense of a [[normed vector space]].)

\begin{defn}
\label{}\hypertarget{}{}
$l^\infty$ is the space of \textbf{absolutely bounded sequences}:

\begin{displaymath}
\sup_k {|a_k|} \lt \infty .
\end{displaymath}
We equip $l^\infty$ with the \textbf{supremum norm}:

\begin{displaymath}
{\|a\|_\infty} \coloneqq \sup_k {|a_k|} .
\end{displaymath}
\end{defn}
This is also a Banach space.

\begin{defn}
\label{}\hypertarget{}{}
$c_c$ (or $c_{00}$) is the space of \textbf{almost-zero sequences}:

\begin{displaymath}
\ess\forall k,\; a_k = 0 ,
\end{displaymath}
where `$\ess\forall$' means `for all but finitely many' ($\tilde{K}$-[[K-finite|finite]] in [[constructive mathematics]]). We equip $c_c$ with the topology of [[compact convergence]] (here, convergence on finite subsets).

\end{defn}
This is a [[locally convex space]].

\begin{defn}
\label{}\hypertarget{}{}
$c_0$ is the space of \textbf{zero-limit sequences}:

\begin{displaymath}
\forall \epsilon,\; \ess\forall k,\; {|a_k|} \lt \epsilon ,
\end{displaymath}
where as usual $\epsilon$ is a [[positive number]] and again `$\ess\forall$' means `for all but finitely many'. We equip $c_0$ with the [[supremum norm]].

\end{defn}
This is also a locally convex space, in fact a [[Banach space]].

\begin{defn}
\label{}\hypertarget{}{}
$c_\infty$ is the space of \textbf{convergent sequences}:

\begin{displaymath}
\exists L,\; \forall \epsilon,\; \ess\forall k,\; {|a_k - L|} \lt \epsilon ,
\end{displaymath}
where $L$ is an element of $K$ and the other notation is as in $c_0$ above. We also equip $c_\infty$ with the supremum norm.

\end{defn}
This is also a [[Banach space]]. $c_\infty$ is also written simply `$c$', but this can be confusing; see the \hyperlink{Generalisations}{Generalisations} below.

There is some argument to be made that an element of $c_\infty$ should be a sequence with the [[extra structure]] of a specific limit $L$, rather than a sequence with the [[extra property]] that some limit exists. This makes no difference if $N$ is [[infinite set|infinite]]; but if $N$ is finite then the version of $c_\infty$ with extra structure is the $l^\infty$-[[l-infinity-direct sum|direct sum]] of the [[ground field]] and the version of $c_\infty$ with extra property.

\begin{defn}
\label{}\hypertarget{}{}
$c_b$ is the space of \textbf{absolutely bounded sequences}:

\begin{displaymath}
\sup_k {|a_k|} \lt \infty .
\end{displaymath}
We equip $c_b$ with the supremum norm too.

\end{defn}
This is yet another [[Banach space]]. Indeed, $c_b = l^\infty$, two different ways of thinking about the same thing. (But they generalise differently.)

\begin{defn}
\label{}\hypertarget{}{}
Finally, $N^K$ is the space of all sequences. We equip $N^K$ with the [[product topology]], also called the topology of [[pointwise convergence]].

\end{defn}
This should probably be denoted `$c$', in line with the \hyperlink{Generalisations}{generalisation} below; but that symbol is often used for $c_\infty$, so it would be confusing.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

These properties all use the version of $c_\infty$ with extra property.

For $0 \lt p \lt q \lt \infty$, we have $c_c \subseteq l^p \subseteq l^q \subseteq c_0$, with each space [[dense subspace|dense]] in the next (using the topology of the next). This continues: $c_0 \subseteq c_\infty \subseteq c_b = l^\infty$, but now each space, far from being dense, is a [[closed subspace]] of the next (with the [[induced topology]]). Finally, $l^\infty = c_b \subseteq K^N$

When $N$ is [[finite set|finite]], these spaces are all the same, being just the [[cartesian spaces]] $K^N$; when $N$ is [[infinite set|infinite]], the inclusions above are all proper (at least if $K$ is nontrivial).

The various [[direct sums of Banach spaces]] follow the sequence spaces $l^p$ for $1 \leq p \leq \infty$.

The [[Riesz representation theorems]] give many nice results for the [[dual vector space|dual spaces]] of the sequence spaces:

\begin{itemize}%
\item the dual of $c_0$ is $l^1$,
\item the dual of $l^p$ is $l^q$ for $p + q = p q$ and $1 \lt p , q \lt \infty$,
\item the dual of $l^1$ is $l^\infty$,
\item the dual of $l^\infty$ is $l^1$ in [[dream mathematics]], but something much larger in [[classical mathematics]].

\end{itemize}
\hypertarget{Generalisations}{}\subsection*{{Generalisations}}\label{Generalisations}

The sequence spaces $l^p$ generalise to the [[Lebesgue spaces]] $L^p$ on arbitrary [[measure spaces]]. In fact, $l^p(N)$ is simply $L^p(N,\mu)$, where $\mu$ is [[counting measure]].

The sequence spaces $c_c$, $c_0$, $c_\infty$, $c_b$, and $K^N$ generalise to the spaces $C_c$, $C_0$, $C_\infty$, $C_b$, and $C$ of [[continuous maps]] on a [[local compactum]]. In fact, $c_*(N)$ is simply $C_*(N,\tau)$, where $\tau$ is the [[discrete topology]]. (Note that one never uses the symbol `$C$' for `$C_\infty$' with capital letters.)

A common setting for both of these generalisations is a (locally compact Hausdorff) [[topological group]]. While $l^\infty$ and $c_b$ are the same, $L^\infty$ and $C_b$ are the same \emph{only} if the group is discrete. (Otherwise $C_b \subset L^\infty$ properly.)

\hypertarget{in_constructive_mathematics}{}\subsection*{{In constructive mathematics}}\label{in_constructive_mathematics}

The spaces $c_c$, $c_0$, and $c_\infty$ work just fine in [[constructive mathematics]] (as does $K^N$, since it has no interesting structure anyway). For $l^p$, we need $N$ to have [[decidable equality]] to define $\|a\|_p$; even so, $\|a\|_p$ (even when bounded) is only a [[lower real number]], so we usually require it be [[located real number|located]] to have an element of $l^p$. With these caveats, $l^p$ works just fine for $0 \lt p \lt \infty$. For $c_b = l^\infty$, we cannot get a [[Banach space]] with [[located real number|located]] [[norms]], as is usually required for constructive [[functional analysis]] \ldots{} well, unless we require $N$ to be [[finite set|finite]] (in the strictest sense), which leaves out the motivating example. Nevertheless, we can still treat $l^\infty$ as a \emph{semicontinuous} Banach space, that is one where the norms may be any bounded [[lower reals]]; for that matter, we can also consider semicontinuous versions of $l^p$. (Another way to treat $l^\infty$ may be formally, as the [[dual vector space|dual]] of $l^1$; I don't know how well this works.)

\hypertarget{references}{}\subsection*{{References}}\label{references}

At least for the term `sequence space', try \href{https://en.wikipedia.org/wiki/Sequence_space}{Wikipedia} and \emph{[[HAF]]}.

[[!redirects sequence space]] [[!redirects sequence spaces]]

[[!redirects l1-space]] [[!redirects l1-spaces]] [[!redirects l-1-space]] [[!redirects l-1-spaces]] [[!redirects l{\tt \symbol{94}}1-space]] [[!redirects l{\tt \symbol{94}}1-spaces]] [[!redirects l\_1-space]] [[!redirects l\_1-spaces]] [[!redirects l1 space]] [[!redirects l1 spaces]] [[!redirects l-1 space]] [[!redirects l-1 spaces]] [[!redirects l{\tt \symbol{94}}1 space]] [[!redirects l{\tt \symbol{94}}1 spaces]] [[!redirects l\_1 space]] [[!redirects l\_1 spaces]]

[[!redirects lp-space]] [[!redirects lp-spaces]] [[!redirects l-p-space]] [[!redirects l-p-spaces]] [[!redirects l{\tt \symbol{94}}p-space]] [[!redirects l{\tt \symbol{94}}p-spaces]] [[!redirects l\_p-space]] [[!redirects l\_p-spaces]] [[!redirects lp space]] [[!redirects lp spaces]] [[!redirects l-p space]] [[!redirects l-p spaces]] [[!redirects l{\tt \symbol{94}}p space]] [[!redirects l{\tt \symbol{94}}p spaces]] [[!redirects l\_p space]] [[!redirects l\_p spaces]]

[[!redirects linfty-space]] [[!redirects linfty-spaces]] [[!redirects l-infty-space]] [[!redirects l-infty-spaces]] [[!redirects l{\tt \symbol{94}}infty-space]] [[!redirects l{\tt \symbol{94}}infty-spaces]] [[!redirects l\_infty-space]] [[!redirects l\_infty-spaces]] [[!redirects linfty space]] [[!redirects linfty spaces]] [[!redirects l-infty space]] [[!redirects l-infty spaces]] [[!redirects l{\tt \symbol{94}}infty space]] [[!redirects l{\tt \symbol{94}}infty spaces]] [[!redirects l\_infty space]] [[!redirects l\_infty spaces]] [[!redirects linfin-space]] [[!redirects linfin-spaces]] [[!redirects l-infin-space]] [[!redirects l-infin-spaces]] [[!redirects l{\tt \symbol{94}}infin-space]] [[!redirects l{\tt \symbol{94}}infin-spaces]] [[!redirects l\_infin-space]] [[!redirects l\_infin-spaces]] [[!redirects linfin space]] [[!redirects linfin spaces]] [[!redirects l-infin space]] [[!redirects l-infin spaces]] [[!redirects l{\tt \symbol{94}}infin space]] [[!redirects l{\tt \symbol{94}}infin spaces]] [[!redirects l\_infin space]] [[!redirects l\_infin spaces]] [[!redirects linfinity-space]] [[!redirects linfinity-spaces]] [[!redirects l-infinity-space]] [[!redirects l-infinity-spaces]] [[!redirects l{\tt \symbol{94}}infinity-space]] [[!redirects l{\tt \symbol{94}}infinity-spaces]] [[!redirects l\_infinity-space]] [[!redirects l\_infinity-spaces]] [[!redirects linfinity space]] [[!redirects linfinity spaces]] [[!redirects l-infinity space]] [[!redirects l-infinity spaces]] [[!redirects l{\tt \symbol{94}}infinity space]] [[!redirects l{\tt \symbol{94}}infinity spaces]] [[!redirects l\_infinity space]] [[!redirects l\_infinity spaces]] [[!redirects l∞-space]] [[!redirects l∞-spaces]] [[!redirects l-∞-space]] [[!redirects l-∞-spaces]] [[!redirects l{\tt \symbol{94}}∞-space]] [[!redirects l{\tt \symbol{94}}∞-spaces]] [[!redirects l\_∞-space]] [[!redirects l\_∞-spaces]] [[!redirects l? space]] [[!redirects l? spaces]] [[!redirects l-∞ space]] [[!redirects l-∞ spaces]] [[!redirects l{\tt \symbol{94}}? space]] [[!redirects l{\tt \symbol{94}}? spaces]] [[!redirects l\_? space]] [[!redirects l\_? spaces]]

[[!redirects cc-space]] [[!redirects cc-spaces]] [[!redirects c-c-space]] [[!redirects c-c-spaces]] [[!redirects c\_c-space]] [[!redirects c\_c-spaces]] [[!redirects cc space]] [[!redirects cc spaces]] [[!redirects c-c space]] [[!redirects c-c spaces]] [[!redirects c\_c space]] [[!redirects c\_c spaces]] [[!redirects c00-space]] [[!redirects c00-spaces]] [[!redirects c-00-space]] [[!redirects c-00-spaces]] [[!redirects c\_00-space]] [[!redirects c\_00-spaces]] [[!redirects c00 space]] [[!redirects c00 spaces]] [[!redirects c-00 space]] [[!redirects c-00 spaces]] [[!redirects c\_00 space]] [[!redirects c\_00 spaces]]

[[!redirects c0-space]] [[!redirects c0-spaces]] [[!redirects c-0-space]] [[!redirects c-0-spaces]] [[!redirects c\_0-space]] [[!redirects c\_0-spaces]] [[!redirects c0 space]] [[!redirects c0 spaces]] [[!redirects c-0 space]] [[!redirects c-0 spaces]] [[!redirects c\_0 space]] [[!redirects c\_0 spaces]]

[[!redirects cinfinity-space]] [[!redirects cinfinity-spaces]] [[!redirects c-infinity-space]] [[!redirects c-infinity-spaces]] [[!redirects c\_infinity-space]] [[!redirects c\_infinity-spaces]] [[!redirects cinfinity space]] [[!redirects cinfinity spaces]] [[!redirects c-infinity space]] [[!redirects c-infinity spaces]] [[!redirects c\_infinity space]] [[!redirects c\_infinity spaces]] [[!redirects cinfin-space]] [[!redirects cinfin-spaces]] [[!redirects c-infin-space]] [[!redirects c-infin-spaces]] [[!redirects c\_infin-space]] [[!redirects c\_infin-spaces]] [[!redirects cinfin space]] [[!redirects cinfin spaces]] [[!redirects c-infin space]] [[!redirects c-infin spaces]] [[!redirects c\_infin space]] [[!redirects c\_infin spaces]] [[!redirects cinfty-space]] [[!redirects cinfty-spaces]] [[!redirects c-infty-space]] [[!redirects c-infty-spaces]] [[!redirects c\_infty-space]] [[!redirects c\_infty-spaces]] [[!redirects cinfty space]] [[!redirects cinfty spaces]] [[!redirects c-infty space]] [[!redirects c-infty spaces]] [[!redirects c\_infty space]] [[!redirects c\_infty spaces]] [[!redirects c∞-space]] [[!redirects c∞-spaces]] [[!redirects c-∞-space]] [[!redirects c-∞-spaces]] [[!redirects c\_∞-space]] [[!redirects c\_∞-spaces]] [[!redirects c? space]] [[!redirects c? spaces]] [[!redirects c-∞ space]] [[!redirects c-∞ spaces]] [[!redirects c\_? space]] [[!redirects c\_? spaces]]

[[!redirects cb-space]] [[!redirects cb-spaces]] [[!redirects c-b-space]] [[!redirects c-b-spaces]] [[!redirects c\_b-space]] [[!redirects c\_b-spaces]] [[!redirects cb space]] [[!redirects cb spaces]] [[!redirects c-b space]] [[!redirects c-b spaces]] [[!redirects c\_b space]] [[!redirects c\_b spaces]]

[[!redirects c-space]] [[!redirects c-spaces]] [[!redirects c space]] [[!redirects c spaces]]



\end{document}