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.

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

For mathematical operations on sequence numbers, see Serial number arithmetic.

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

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

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

${\displaystyle (x_{n})_{n\in \mathbf {N} }+(y_{n})_{n\in \mathbf {N} }{\stackrel {\rm {def}}{=}}(x_{n}+y_{n})_{n\in \mathbf {N} }}$

and the scalar multiplication as

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

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

${\displaystyle \sum _{n}|x_{n}|^{p}<\infty .}$

If p ≥ 1, then the real-valued operation ${\displaystyle \|\cdot \|_{p}}$ defined by

${\displaystyle \|x\|_{p}=\left(\sum _{n}|x_{n}|^{p}\right)^{1/p}}$

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

${\displaystyle d(x,y)=\sum _{n}|x_{n}-y_{n}|^{p}.\,}$

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

${\displaystyle \|x\|_{\infty }=\sup _{n}|x_{n}|,}$

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. This is not a closed subspace and therefore is not a Banach space with respect to the infinity norm. For example, the sequence (x_n) where x_n has 1/n for the first n entries and is zero everywhere else (i.e. x_n = (1, 1/2, ..., 1/n, 0, ...)) is Cauchy w.r.t. infinity norm but not convergent (to a sequence in c₀₀).

Other sequence spaces

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

${\displaystyle \sup _{n}\left\vert \sum _{i=0}^{n}x_{i}\right\vert <\infty .}$

This space, when equipped with the norm

${\displaystyle \|x\|_{bs}=\sup _{n}\left\vert \sum _{i=0}^{n}x_{i}\right\vert ,}$

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

${\displaystyle (x_{n})_{n\in \mathbf {N} }\mapsto \left(\sum _{i=0}^{n}x_{i}\right)_{n\in \mathbf {N} }.}$

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 ${\displaystyle c_{00}}$ 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

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

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

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

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

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

so that the operator norm satisfies

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

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

${\displaystyle y_{n}={\begin{cases}0&{\rm {{if}\ x_{n}=0}}\\x_{n}^{-1}|x_{n}|^{q}&{\rm {{if}\ x_{n}\not =0}}\end{cases}}}$

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

${\displaystyle \|L_{x}\|_{(\ell ^{p})^{*}}=\|x\|_{q}.}$

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 ${\displaystyle x\mapsto L_{x}}$ gives an isometry

${\displaystyle \kappa _{q}:\ell ^{q}\to (\ell ^{p})^{*}.}$

The map

${\displaystyle \ell ^{q}{\xrightarrow {\kappa _{q}}}(\ell ^{p})^{*}{\xrightarrow {(\kappa _{q}^{*})^{-1}}}}$

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 ${\displaystyle Q:\ell ^{1}\to X}$, so that X is isomorphic to ${\displaystyle \ell ^{1}/\ker Q}$. In general, ker Q is not complemented in ℓ¹, that is, there does not exist a subspace Y of ℓ¹ such that ${\displaystyle \ell ^{1}=Y\oplus \ker Q}$. In fact, ℓ¹ has uncountably many uncomplemented subspaces that are not isomorphic to one another (for example, take ${\displaystyle X=\ell ^{p}}$; 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 ${\displaystyle p\in [1,\infty ]}$, the spaces ${\displaystyle \ell ^{p}}$ are increasing in ${\displaystyle p}$, with the inclusion operator being continuous: for ${\displaystyle 1\leq p<q\leq \infty }$, one has ${\displaystyle \|f\|_{q}\leq \|f\|_{p}}$.

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

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.[1] If K is a subset of this space, then the following are equivalent:[1]

1. K is compact;
2. K is weakly compact;
3. K is bounded, closed, and equismall at infinity.

Here K being equismall at infinity means that for every ${\displaystyle \epsilon >0}$, there exists an integer ${\displaystyle n_{\epsilon }\geq 0}$ such that ${\displaystyle \sum _{n=n_{\epsilon }}^{\infty }|s_{n}|<\epsilon }$ for all ${\displaystyle s=\left(s_{n}\right)_{n=1}^{\infty }\in K}$.

See also

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

References

• Banach, Stefan; 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.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)
• 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.
• Trèves, François (August 6, 2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (link) CS1 maint: date and year (link)

1. Treves 2006, pp. 451-458. sfn error: no target: CITEREFTreves2006 (help)