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 K 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 K, 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 ℓ^p spaces, consisting of the p-power summable sequences, with the p-norm. These are special cases of L^p 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

Let K denote the field either of real or complex numbers. Denote by K^N the set of all sequences of scalars

$(x_n)_{n\in\mathbf{N}},\quad x_n\in\mathbf{K}.$

This can be turned into a vector space by defining vector addition as

$(x_n)_{n\in\mathbf{N}} + (y_n)_{n\in\mathbf{N}} \stackrel{\rm{def}}{=} (x_n + y_n)_{n\in\mathbf{N}}$

and the scalar multiplication as

$\alpha(x_n)_{n\in\mathbf{N}} := (\alpha x_n)_{n\in\mathbf{N}}.$

A sequence space is any linear subspace of K^N.

ℓ^p spaces

For 0 < p < ∞, ℓ^p is the subspace of K^N consisting of all sequences x = (x_n) satisfying

$\sum_n |x_n|^p < \infty.$

If p ≥ 1, then the real-valued operation $\|\cdot\|_p$ defined by

$\|x\|_p = \left(\sum_n|x_n|^p\right)^{1/p}$

defines a norm on ℓ^p. In fact, ℓ^p is a complete metric space with respect to this norm, and therefore is a Banach space.

If 0 < p < 1, then ℓ^p does not carry a norm, but rather a metric defined by

$d(x,y) = \sum_n |x_n-y_n|^p.\,$

If p = ∞, then ℓ^∞ is defined to be the space of all bounded sequences. With respect to the norm

$\|x\|_\infty = \sup_n |x_n|,$

ℓ^∞ is also a Banach space.

c and c₀

The space of convergent sequences c is a sequence space. This consists of all x ∈ K^N such that lim_n→∞x_n exists. Since every convergent sequence is bounded, c is a linear subspace of ℓ^∞. It is, moreover, a closed subspace with respect to the infinity norm, and so a Banach space in its own right.

The subspace of null sequences c₀ consists of all sequences whose limit is zero. This is a closed subspace of c, and so again a Banach space.

Other sequence spaces

The space of bounded series, denote by bs, is the space of sequences x for which

$\sup_n \left\vert \sum_{i=0}^n x_i \right\vert < \infty.$

This space, when equipped with the norm

$\|x\|_{bs} = \sup_n \left\vert \sum_{i=0}^n x_i \right\vert,$

is a Banach space isometrically isomorphic to ℓ^∞, via the linear mapping

$(x_n)_{n\in\mathbf{N}} \mapsto \left(\sum_{i=0}^n x_i\right)_{n\in\mathbf{N}}.$

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 $c_{00}$ 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.

Properties of ℓ^p spaces and the space c₀

The space ℓ² is the only ℓ^p space that is a Hilbert space, since any norm that is induced by an inner product should satisfy the parallelogram identity $\|x+y\|_p^2 + \|x-y\|_p^2= 2\|x\|_p^2 + 2\|y\|_p^2$. Substituting two distinct unit vectors for x and y directly shows that the identity is not true unless p = 2.

Each ℓ^p is distinct, in that ℓ^p is a strict subset of ℓ^s whenever p < s; furthermore, ℓ^p is not linearly isomorphic to ℓ^s when p ≠ s. In fact, by Pitt's theorem (Pitt 1936), every bounded linear operator from ℓ^s to ℓ^p 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 ℓ^s, and is thus said to be strictly singular.

If 1 < p < ∞, then the (continuous) dual space of ℓ^p is isometrically isomorphic to ℓ^q, where q is the Hölder conjugate of p: 1/p + 1/q = 1. The specific isomorphism associates to an element x of ℓ^q the functional

$L_x(y) = \sum_n x_ny_n$

for y in ℓ^p. Hölder's inequality implies that L_x is a bounded linear functional on ℓ^p, and in fact

$|L_x(y)| \le \|x\|_q\,\|y\|_p$

so that the operator norm satisfies

$\|L_x\|_{(\ell^p)^*} \stackrel{\rm{def}}{=}\sup_{y\in\ell^p, y\not=0} \frac{|L_x(y)|}{\|y\|_p} \le \|x\|_q.$

In fact, taking y to be the element of ℓ^p with

$y_n = \begin{cases}0&\rm{if}\ x_n=0\\ x_n^{-1}|x_n|^q &\rm{if}\ x_n\not=0 \end{cases}$

gives L_x(y) = ||x||_q, so that in fact

$\|L_x\|_{(\ell^p)^*} = \|x\|_q.$

Conversely, given a bounded linear functional L on ℓ^p, the sequence defined by x_n = L(e_n) lies in ℓ^q. Thus the mapping $x\mapsto L_x$ gives an isometry

$\kappa_q : \ell^q \to (\ell^p)^*.$

The map

$\ell^q\xrightarrow{\kappa_q}(\ell^p)^*\xrightarrow{(\kappa_q^*)^{-1}}$

obtained by composing κ_p with the inverse of its transpose coincides with the canonical injection of ℓ^q into its double dual. As a consequence ℓ^q is a reflexive space. By abuse of notation, it is typical to identify ℓ^q with the dual of ℓ^p: (ℓ^p)^* = ℓ^q. Then reflexivity is understood by the sequence of identifications (ℓ^p)^** = (ℓ^q)^* = ℓ^p.

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 ℓ^p and c₀ are separable, with the sole exception of ℓ^∞. The dual of ℓ^∞ is the ba space.

The spaces c₀ and ℓ^p (for 1 ≤ p < ∞) have a canonical unconditional Schauder basis {e_i | i = 1, 2,…}, where e_i is the sequence which is zero but for a 1 in the i^th entry.

The space ℓ¹ has the Schur property: In ℓ¹, any sequence that is weakly convergent is also strongly convergent (Schur 1921). 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 ℓ^p spaces can be embedded into many Banach spaces. The question of whether every infinite-dimensional Banach space contains an isomorph of some ℓ^p 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 $Q:\ell^1 \to X$, so that X is isomorphic to $\ell^1 / \ker Q$. In general, ker Q is not complemented in ℓ¹, that is, there does not exist a subspace Y of ℓ¹ such that $\ell^1 = Y \oplus \ker Q$. In fact, ℓ¹ has uncountably many uncomplemented subspaces that are not isomorphic to one another (for example, take $X=\ell^p$; since there are uncountably many such X 's, and since no ℓ^p is isomorphic to any other, there are thus uncountably many ker Q 's).

Except for the trivial finite-dimensional case, an unusual feature of ℓ^p is that it is not polynomially reflexive.

ℓ^p spaces are increasing in p

For $p\in[1,+\infty]$, the spaces $\ell^p$ are increasing in $p$, with $1\le p<q\le+\infty$ implying $\|f\|_q\le\|f\|_p$.

This follows from defining $F:=\frac{f}{\|f\|_p}$ for $f\in\ell^p$, and noting that $|F(m)|\le1$ for all $m\in\mathbb{N}$, which can be shown to imply $\|F\|_q^q\le1$.

See also

• L^p space
• Tsirelson space
• beta-dual space

References

• Banach, S.; Mazur, S. (1933), "Zur Theorie der linearen Dimension", Studia Mathematica 4: 100–112.
• Dunford, Nelson; Schwartz, Jacob T. (1958), Linear operators, volume I, Wiley-Interscience.
• Pitt, H.R. (1936), "A note on bilinear forms", J. London Math. Soc. 11 (3): 174–180, doi:10.1112/jlms/s1-11.3.174.
• 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.