nLab
sequence space

Sequence spaces

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 classical sequence spaces are spaces of infinite sequences (hence the name), although they work for functions on any set.

Terminology

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.

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 $ℂ$ of complex numbers or the space $ℝ$ 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 $K$-valued $N$-sequence, or simply a 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.

Definition

$l^1$ is the space of absolutely summable sequences:

$$\sum _k\mid a_k\mid <\mathrm{\infty }.$$\sum_k {|a_k|} \lt \infty .

We equip $l^1$ with the $l^1$-norm

$$\parallel a{\parallel }_1≔\sum _k\mid a_k\mid .$${\|a\|_1} \coloneqq \sum_k {|a_k|} .

This is a Banach space.

Definition

$l^2$ is the space of absolutely square-summable sequences (or, over a real field?, simply square-summable sequences):

$$\sum _k\mid a_k{\mid }^2<\mathrm{\infty }.$$\sum_k {|a_k|^2} \lt \infty .

We equip $l^2$ with the $l^2$-norm

$$\parallel a{\parallel }_2≔\sqrt{\sum _k\mid a_k{\mid }^2}.$${\|a\|_2} \coloneqq \sqrt{\sum_k {|a_k|^2}} .

This is also a Banach space; in fact, it's a Hilbert space (assuming that $K$ is). Furthermore, every Hilbert space (over $K$ a field) arises in this way, up to isometric isomorphism?, using an orthonormal basis for $N$.

More generally, for $0<p<\mathrm{\infty }$:

Definition

$l^p$ is the space of absolutely $p$th-power–summable sequences:

$$\sum _k\mid a_k{\mid }^p<\mathrm{\infty }.$$\sum_k {|a_k|^p} \lt \infty .

We equip $l^p$ with the $l^p$-norm

$$\parallel a{\parallel }_p≔\sqrt[p]{\sum _k\mid a_k{\mid }^p}.$${\|a\|_p} \coloneqq \sqrt[p]{\sum_k {|a_k|^p}} .

This is at least an $F$-space?, which is a Banach space iff $p\ge 1$. (For $p<1$, the ‘norm’ is not really a norm in the sense of a normed vector space.)

Definition

$l^{\mathrm{\infty }}$ is the space of absolutely bounded sequences:

$$\underset{k}{\sup }\mid a_k\mid <\mathrm{\infty }.$$\sup_k {|a_k|} \lt \infty .

We equip $l^{\mathrm{\infty }}$ with the supremum norm:

$$\parallel a{\parallel }_{\mathrm{\infty }}≔\underset{k}{\sup }\mid a_k\mid .$${\|a\|_\infty} \coloneqq \sup_k {|a_k|} .

This is also a Banach space.

Definition

$c_c$ (or $c_{00}$) is the space of almost-zero sequences:

$$ess\forall k,a_k=0,$$\ess\forall k,\; a_k = 0 ,

where ‘$ess\forall$’ means ‘for all but finitely many’ ($\tilde{K}$-finite in constructive mathematics). We equip $c_c$ with the topology of compact convergence? (here, convergence on finite subsets).

This is a locally convex space.

Definition

$c_0$ is the space of zero-limit sequences:

$$\forall ϵ,ess\forall k,\mid a_k\mid <ϵ,$$\forall \epsilon,\; \ess\forall k,\; {|a_k|} \lt \epsilon ,

where as usual $ϵ$ is a positive number and again ‘$ess\forall$’ means ‘for all but finitely many’. We equip $c_0$ with the supremum norm?.

This is also a locally convex space, in fact a Banach space.

Definition

$c_{\mathrm{\infty }}$ is the space of convergent sequences:

$$\exists L,\forall ϵ,ess\forall k,\mid a_k-L\mid <ϵ,$$\exists L,\; \forall \epsilon,\; \ess\forall k,\; {|a_k - L|} \lt \epsilon ,

where $L$ is an element of $K$ and the other notation is as in $c_0$ above. We also equip $c_{\mathrm{\infty }}$ with the supremum norm.

This is also a Banach space. $c_{\mathrm{\infty }}$ is also written simply ‘$c$’, but this can be confusing; see the Generalisations below.

There is some argument to be made that an element of $c_{\mathrm{\infty }}$ 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; but if $N$ is finite then the version of $c_{\mathrm{\infty }}$ with extra structure is the $l^{\mathrm{\infty }}$-direct sum of the ground field and the version of $c_{\mathrm{\infty }}$ with extra property.

Definition

$c_b$ is the space of absolutely bounded sequences:

$$\underset{k}{\sup }\mid a_k\mid <\mathrm{\infty }.$$\sup_k {|a_k|} \lt \infty .

We equip $c_b$ with the supremum norm too.

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

Definition

Finally, $N^K$ is the space of all sequences. We equip $N^K$ with the product topology, also called the topology of pointwise convergence?.

This should probably be denoted ‘$c$’, in line with the generalisation below; but that symbol is often used for $c_{\mathrm{\infty }}$, so it would be confusing.

Properties

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

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

When $N$ is finite, these spaces are all the same, being just the cartesian spaces $K^N$; when $N$ is 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 $1\le p\le \mathrm{\infty }$.

The Riesz representation theorems give many nice results for the dual spaces of the sequence spaces:

• the dual of $c_0$ is $l^1$,
• the dual of $l^p$ is $l^q$ for $p+q=pq$ and $1<p,q<\mathrm{\infty }$,
• the dual of $l^1$ is $l^{\mathrm{\infty }}$,
• the dual of $l^{\mathrm{\infty }}$ is $l^1$ in dream mathematics, but something much larger in classical mathematics.

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 $c_c$, $c_0$, $c_{\mathrm{\infty }}$, $c_b$, and $K^N$ generalise to the spaces $C_c$, $C_0$, $C_{\mathrm{\infty }}$, $C_b$, 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 ‘$C_{\mathrm{\infty }}$’ with capital letters.)

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

In constructive mathematics

The spaces $c_c$, $c_0$, and $c_{\mathrm{\infty }}$ 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 $\parallel a{\parallel }_p$; even so, $\parallel a{\parallel }_p$ (even when bounded) is only a lower real number, so we usually require it be located to have an element of $l^p$. With these caveats, $l^p$ works just fine for $0<p<\mathrm{\infty }$. For $c_b=l^{\mathrm{\infty }}$, we cannot get a Banach space with located norms, as is usually required for constructive functional analysis … well, unless we require $N$ to be finite (in the strictest sense), which leaves out the motivating example. Nevertheless, we can still treat $l^{\mathrm{\infty }}$ as a 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 $l^{\mathrm{\infty }}$ may be formally, as the dual of $l^1$; I don't know how well this works.)

References

At least for the term ‘sequence space’, try Wikipedia and HAF.