nLab
sequence space

Context

Functional analysis

Topology

Sequence spaces

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 0c_0’ and ‘l pl^p’, that appear below. The term ‘sequence space’ is useful as a general name without symbols in it.

Definitions

Fix a set NN; typically, NN 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 NN be a finite set.

Also fix a topological vector space KK; typically, KK is either the space \mathbb{C} of complex numbers or the space \mathbb{R} of real numbers. We will assume below that KK 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 KK.

We will think of a function from NN to KK as a KK-valued NN-sequence, or simply a sequence. The various sequence spaces will be subsets of the function set K NK^N of all sequences. In general, if ‘XX’ is the symbol for a sequence space, then we may specify NN and KK by writing ‘X(N,K)X(N,K)’ (or a variation thereon), but often this is suppressed.

Definition

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

k|a k|<. \sum_k {|a_k|} \lt \infty .

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

a 1 k|a k|. {\|a\|_1} \coloneqq \sum_k {|a_k|} .

This is a Banach space.

Definition

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

k|a k| 2<. \sum_k {|a_k|^2} \lt \infty .

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

a 2 k|a k| 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 KK is). Furthermore, every Hilbert space (over KK a field) arises in this way, up to isometric isomorphism?, using an orthonormal basis for NN.

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

Definition

l pl^p is the space of absolutely ppth-power–summable sequences:

k|a k| p<. \sum_k {|a_k|^p} \lt \infty .

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

a p k|a k| pp. {\|a\|_p} \coloneqq \sqrt[p]{\sum_k {|a_k|^p}} .

This is at least an FF-space?, which is a Banach space iff p1p \geq 1. (For p<1p \lt 1, the ‘norm’ is not really a norm in the sense of a normed vector space.)

Definition

l l^\infty is the space of absolutely bounded sequences:

sup k|a k|<. \sup_k {|a_k|} \lt \infty .

We equip l l^\infty with the supremum norm:

a sup k|a k|. {\|a\|_\infty} \coloneqq \sup_k {|a_k|} .

This is also a Banach space.

Definition

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

essk,a k=0, \ess\forall k,\; a_k = 0 ,

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

This is a locally convex space.

Definition

c 0c_0 is the space of zero-limit sequences:

ϵ,essk,|a k|<ϵ, \forall \epsilon,\; \ess\forall k,\; {|a_k|} \lt \epsilon ,

where as usual ϵ\epsilon is a positive number and again ‘ess\ess\forall’ means ‘for all but finitely many’. We equip c 0c_0 with the supremum norm?.

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

Definition

c c_\infty is the space of convergent sequences:

L,ϵ,essk,|a kL|<ϵ, \exists L,\; \forall \epsilon,\; \ess\forall k,\; {|a_k - L|} \lt \epsilon ,

where LL is an element of KK and the other notation is as in c 0c_0 above. We also equip c c_\infty with the supremum norm.

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

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

Definition

c bc_b is the space of absolutely bounded sequences:

sup k|a k|<. \sup_k {|a_k|} \lt \infty .

We equip c bc_b with the supremum norm too.

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

Definition

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

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

Properties

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

For 0<p<q<0 \lt p \lt q \lt \infty, we have c cl pl qc 0c_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 0c c b=l 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 =c bK Nl^\infty = c_b \subseteq K^N

When NN is finite, these spaces are all the same, being just the cartesian spaces K NK^N; when NN is infinite, the inclusions above are all proper (at least if KK is nontrivial).

The various direct sums of Banach spaces follow the sequence spaces l pl^p for 1p1 \leq p \leq \infty.

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

  • the dual of c 0c_0 is l 1l^1,
  • the dual of l pl^p is l ql^q for p+q=pqp + q = p q and 1<p,q<1 \lt p , q \lt \infty,
  • the dual of l 1l^1 is l l^\infty,
  • the dual of l l^\infty is l 1l^1 in dream mathematics, but something much larger in classical mathematics.

Generalisations

The sequence spaces l pl^p generalise to the Lebesgue spaces L pL^p on arbitrary measure spaces. In fact, l p(N)l^p(N) is simply L p(N,μ)L^p(N,\mu), where μ\mu is counting measure?.

The sequence spaces c cc_c, c 0c_0, c c_\infty, c bc_b, and K NK^N generalise to the spaces C cC_c, C 0C_0, C C_\infty, C bC_b, and CC of continuous maps on a local compactum. In fact, c *(N)c_*(N) is simply C *(N,τ)C_*(N,\tau), where τ\tau is the discrete topology. (Note that one never uses the symbol ‘CC’ for ‘C C_\infty’ with capital letters.)

A common setting for both of these generalisations is a (locally compact Hausdorff) topological group. While l l^\infty and c bc_b are the same, L L^\infty and C bC_b are the same only if the group is discrete. (Otherwise C bL C_b \subset L^\infty properly.)

In constructive mathematics

The spaces c cc_c, c 0c_0, and c c_\infty work just fine in constructive mathematics (as does K NK^N, since it has no interesting structure anyway). For l pl^p, we need NN to have decidable equality to define a p\|a\|_p; even so, a p\|a\|_p (even when bounded) is only a lower real number, so we usually require it be located to have an element of l pl^p. With these caveats, l pl^p works just fine for 0<p<0 \lt p \lt \infty. For c b=l c_b = l^\infty, we cannot get a Banach space with located norms, as is usually required for constructive functional analysis … well, unless we require NN to be finite (in the strictest sense), which leaves out the motivating example. Nevertheless, we can still treat l l^\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 pl^p. (Another way to treat l l^\infty may be formally, as the dual of l 1l^1; I don't know how well this works.)

References

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

Revised on August 10, 2014 21:23:24 by Toby Bartels (98.16.175.66)