Sequence space
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.
The most important sequence spaces in analysis are the spaces, consisting of the -power summable sequences, with the -norm. These are special cases of spaces for the counting measure on the set of natural numbers. Other important classes of sequences like convergent sequences or [|null sequences] form sequence spaces, respectively denoted and, with the sup norm. Any sequence space can also be equipped with the topology of pointwise convergence, under which it becomes a special kind of Fréchet space called FK-space.
Definition
A sequence in a set is an -valued map whose value at is denoted by instead of the usual parentheses notation.Space of all sequences
Let denote the field either of real or complex numbers. The set of all sequences of elements of is a vector space for componentwise additionand componentwise scalar multiplication
A sequence space is any linear subspace of.
As a topological space, is naturally endowed with the product topology. Under this topology, is Fréchet, meaning that it is a complete, metrizable, locally convex topological vector space. However, this topology is rather pathological: there are no continuous norms on . Among Fréchet spaces, is minimal in having no continuous norms:
But the product topology is also unavoidable: does not admit a strictly coarser Hausdorff, locally convex topology. For that reason, the study of sequences begins by finding a strict linear subspace of interest, and endowing it with a topology different from the subspace topology.
spaces
For, is the subspace of consisting of all sequences satisfyingIf, then the real-valued function on defined by
defines a norm on. In fact, is a complete metric space with respect to this norm, and therefore is a Banach space.
If then is also a Hilbert space when endowed with its canonical inner product, called the , defined for all by
The canonical norm induced by this inner product is the usual -norm, meaning that for all.
If, then is defined to be the space of all bounded sequences endowed with the norm
is also a Banach space.
If, then does not carry a norm, but rather a metric defined by
, and
A is any sequence such that exists.The set of all convergent sequences is a vector subspace of called the c space|. Since every convergent sequence is bounded, is a linear subspace of. Moreover, this sequence space is a closed subspace of with respect to the supremum norm, and so it is a Banach space with respect to this norm.
A sequence that converges to is called a and is said to. The set of all sequences that converge to is a closed vector subspace of that when endowed with the supremum norm becomes a Banach space that is denoted by and is called the or the.
The,, is the subspace of consisting of all sequences which have only finitely many nonzero elements. This is not a closed subspace and therefore is not a Banach space with respect to the infinity norm. For example, the sequence where for the first entries and is zero everywhere else is a Cauchy sequence but it does not converge to a sequence in
Space of all finite sequences
Letdenote the space of finite sequences over. As a vector space, is equal to, but has a different topology.
For every natural number, let denote the usual Euclidean space endowed with the Euclidean topology and let denote the canonical inclusion
The image of each inclusion is
and consequently,
This family of inclusions gives a final topology, defined to be the finest topology on such that all the inclusions are continuous. With this topology, becomes a complete, Hausdorff, locally convex, sequential, topological vector space that is Fréchet–Urysohn. The topology is also strictly finer than the subspace topology induced on by.
Convergence in has a natural description: if and is a sequence in then in if and only is eventually contained in a single image and under the natural topology of that image.
Often, each image is identified with the corresponding ; explicitly, the elements and are identified. This is facilitated by the fact that the subspace topology on the quotient topology from the map and the Euclidean topology on all coincide. With this identification, is the direct limit of the directed system where every inclusion adds trailing zeros:
This shows is an LB-space.
Other sequence spaces
The space of bounded series, denote by bs, is the space of sequences for whichThis space, when equipped with the norm
is a Banach space isometrically isomorphic to via the linear mapping
The subspace consisting of all convergent series is a subspace that goes over to the space under this isomorphism.
The space or is defined to be the space of all infinite sequences with only a finite number of non-zero terms. This set is dense in many sequence spaces.
Properties of spaces and the space
The space is the only space that is a Hilbert space, since any norm that is induced by an inner product should satisfy the parallelogram lawSubstituting two distinct unit vectors for and directly shows that the identity is not true unless.
Each is distinct, in that is a strict subset of whenever ; furthermore, is not linearly isomorphic to when. In fact, by Pitt's theorem, every bounded linear operator from to is compact when. No such operator can be an isomorphism; and further, it cannot be an isomorphism on any infinite-dimensional subspace of, and is thus said to be strictly singular.
If, then the dual space of is isometrically isomorphic to, where is the Hölder conjugate of :. The specific isomorphism associates to an element of the functional
for in. Hölder's inequality implies that is a bounded linear functional on, and in fact
so that the operator norm satisfies
In fact, taking to be the element of with
gives so that in fact
Conversely, given a bounded linear functional on, the sequence defined by lies in. Thus the mapping gives an isometry
The map
obtained by composing with the inverse of its transpose coincides with the canonical injection of into its double dual. As a consequence is a reflexive space. By abuse of notation, it is typical to identify with the dual of :. Then reflexivity is understood by the sequence of identifications.
The space is defined as the space of all sequences converging to zero, with norm identical to It is a closed subspace of, hence a Banach space. The dual of is ; the dual of is. For the case of natural numbers index set, the and are separable, with the sole exception of. The dual of is the ba space.
The spaces and have a canonical unconditional Schauder basis, where is the sequence which is zero but for a in the th entry.
The space ℓ1 has the Schur property: In ℓ1, any sequence that is weakly convergent is also strongly convergent. However, since the weak topology on infinite-dimensional spaces is strictly weaker than the strong topology, there are nets in ℓ1 that are weak convergent but not strong convergent.
The spaces can be embedded into many Banach spaces. The question of whether every infinite-dimensional Banach space contains an isomorph of some or of, was answered negatively by B. S. Tsirelson's construction of Tsirelson space in 1974. The dual statement, that every separable Banach space is linearly isometric to a quotient space of, was answered in the affirmative by. That is, for every separable Banach space, there exists a quotient map, so that is isomorphic to. In general, is not complemented in, that is, there does not exist a subspace of such that. In fact, has uncountably many uncomplemented subspaces that are not isomorphic to one another.
Except for the trivial finite-dimensional case, an unusual feature of is that it is not polynomially reflexive.
spaces are increasing in
For, the spaces are increasing in, with the inclusion operator being continuous: for, one has Indeed, the inequality is homogeneous in the, so it is sufficient to prove it under the assumption that In this case, we need only show that for. But if then for all, and thenis isomorphic to all separable, infinite dimensional Hilbert spaces
Let be a separable Hilbert space. Every orthogonal set in is at most countable. The following two items are related:- If is infinite dimensional, then it is isomorphic to,
- If, then is isomorphic to.
Properties of spaces
If is a subset of this space, then the following are equivalent:
- is compact;
- is weakly compact;
- is bounded, closed, and equismall at infinity.