Sequence space

For usage in evolutionary biology, see Sequence space (evolution).

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

See also: L^p space and L-infinity

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, c₀ and c₀₀

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.

The subspace of eventually zero sequences c₀₀ consists of all sequences which have only finitely many nonzero elements.

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

See also: c space

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 law

$\|x+y\|_{p}^{2}+\|x-y\|_{p}^{2}=2\|x\|_{p}^{2}+2\|y\|_{p}^{2}.$

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

$L_{x}(y)=\sum _{n}x_{n}y_{n}$

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

$|L_{x}(y)|\leq \|x\|_{q}\,\|y\|_{p}$

so that the operator norm satisfies

$\|L_{x}\|_{(\ell ^{p})^{*}}{\stackrel {\rm {def}}{=}}\sup _{y\in \ell ^{p},y\not =0}{\frac {|L_{x}(y)|}{\|y\|_{p}}}\leq \|x\|_{q}.$

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 the inclusion operator being continuous: for $1\leq p<q\leq +\infty$, one has $\|f\|_{q}\leq \|f\|_{p}$.

This follows from defining $F:={\frac {f}{\|f\|_{p}}}$ for $f\in \ell ^{p}$, and noting that $|F(m)|\leq 1$ for all $m\in \mathbb {N}$, which can be shown to imply $\|F\|_{q}^{q}\leq 1$.

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.