The most important sequence spaces in analysis are the ℓp spaces, consisting of the p-power summable sequences, with the p-norm. These are special cases of
Lp 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 c and c0, 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 just 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 addition
Theorem[1] — Let be a
Fréchet space over
Then the following are equivalent:
admits no continuous norm (that is, any continuous seminorm on has a nontrivial null space).
contains a vector subspace TVS-isomorphic to .
contains a complemented vector subspace TVS-isomorphic to .
But the product topology is also unavoidable: does not admit a
strictly coarser Hausdorff, locally convex topology.[1] 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.
A convergent sequence is any sequence such that exists.
The set of all convergent sequences is a vector subspace of called the
space of convergent sequences. 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 null sequence and is said to vanish. 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 space of null sequences or the space of vanishing sequences.
The space of eventually zero sequences, 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 (for ) and is zero everywhere else (that is, ) is a
Cauchy sequence but it does not converge to a sequence in
Space of all finite sequences
Let
,
denote the space of finite sequences over. As a vector space, is equal to , but has a different topology.
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:
The space of bounded
series, denote by
bs, is the space of sequences for which
This space, when equipped with the norm
is a Banach space isometrically isomorphic to via the
linear mapping
The subspace cs consisting of all convergent series is a subspace that goes over to the space c under this isomorphism.
The space Φ or is defined to be the space of all infinite sequences with only a finite number of non-zero terms (sequences with
finite support). This set is
dense in many sequence spaces.
Substituting two distinct unit vectors for x and y directly shows that the identity is not true unless p = 2.
Each ℓp is distinct, in that ℓp is a strict
subset of ℓs whenever p < s; furthermore, ℓp is not linearly
isomorphic to ℓs when p ≠ s. In fact, by Pitt's theorem (
Pitt 1936), every bounded linear operator from ℓs to ℓp is
compact when p < s. No such operator can be an isomorphism; and further, it cannot be an isomorphism on any infinite-dimensional subspace of ℓs, and is thus said to be
strictly singular.
If 1 < p < ∞, then the
(continuous) dual space of ℓp is isometrically isomorphic to ℓq, where q is the
Hölder conjugate of p: 1/p + 1/q = 1. The specific isomorphism associates to an element x of ℓq the functional
for y in ℓp.
Hölder's inequality implies that Lx is a bounded linear functional on ℓp, and in fact
so that the operator norm satisfies
In fact, taking y to be the element of ℓp with
gives Lx(y) = ||x||q, so that in fact
Conversely, given a bounded linear functional L on ℓp, the sequence defined by xn = L(en) lies in ℓq. Thus the mapping gives an isometry
The map
obtained by composing κp with the inverse of its
transpose coincides with the
canonical injection of ℓq into its
double dual. As a consequence ℓq is a
reflexive space. By
abuse of notation, it is typical to identify ℓq with the dual of ℓp: (ℓp)* = ℓq. Then reflexivity is understood by the sequence of identifications (ℓp)** = (ℓq)* = ℓp.
The space c0 is defined as the space of all sequences converging to zero, with norm identical to ||x||∞. It is a closed subspace of ℓ∞, hence a Banach space. The
dual of c0 is ℓ1; the dual of ℓ1 is ℓ∞. For the case of natural numbers index set, the ℓp and c0 are
separable, with the sole exception of ℓ∞. The dual of ℓ∞ is the
ba space.
The spaces c0 and ℓp (for 1 ≤ p < ∞) have a canonical unconditional
Schauder basis {ei | i = 1, 2,...}, where ei is the sequence which is zero but for a 1 in the i th entry.
The ℓp spaces can be
embedded into many
Banach spaces. The question of whether every infinite-dimensional Banach space contains an isomorph of some ℓp or of c0, 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 ℓ1, was answered in the affirmative by
Banach & Mazur (1933). That is, for every separable Banach space X, there exists a quotient map , so that X is isomorphic to . In general, ker Q is not complemented in ℓ1, that is, there does not exist a subspace Y of ℓ1 such that . In fact, ℓ1 has uncountably many uncomplemented subspaces that are not isomorphic to one another (for example, take ; since there are uncountably many such X's, and since no ℓp is isomorphic to any other, there are thus uncountably many ker Q's).
Except for the trivial finite-dimensional case, an unusual feature of ℓp is that it is not
polynomially reflexive.
ℓp spaces are increasing in p
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 then .
ℓ2 is isomorphic to all separable, infinite dimensional Hilbert spaces
Let H be a
separable Hilbert space. Every orthogonal set in H is at most countable (i.e. has finite
dimension or ).[2] The following two items are related:
If H is infinite dimensional, then it is isomorphic to ℓ2
If dim(H) = N, then H is isomorphic to
Properties of ℓ1 spaces
A sequence of elements in ℓ1 converges in the space of complex sequences ℓ1 if and only if it converges weakly in this space.[3]
If K is a subset of this space, then the following are equivalent:[3]
K is compact;
K is weakly compact;
K is bounded, closed, and equismall at infinity.
Here K being equismall at infinity means that for every , there exists a natural number such that for all .
Schur, J. (1921), "Über lineare Transformationen in der Theorie der unendlichen Reihen", Journal für die reine und angewandte Mathematik, 151: 79–111,
doi:
10.1515/crll.1921.151.79.