|
|
| Line 1: |
Line 1: |
| {{for|usage in evolutionary biology|Sequence space (evolution)}}
| | Greetings. I want to start by telling you the author's name - Ela Vitela. Connecticut is just place I have been residing in but I will have to maneuver in yearly or two. My job is a cashier. Greeting card collecting is one item he loves most. My husband and I maintain a website. It's advisable to do it here: http://www.pinterest.com/seodress/granite-bay-real-estate-expert-realtor-in-all-thin/ |
| In [[functional analysis]] and related areas of [[mathematics]], a '''sequence space''' is a [[vector space]] whose elements are infinite [[sequence]]s of [[real number|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 sequence]]s 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 subspace]]s of this space. Sequence spaces are typically equipped with a [[norm (mathematics)|norm]], or at least the structure of a [[topological vector space]].
| |
| | |
| The most important sequences spaces in analysis are the ℓ<sup>''p''</sup> spaces, consisting of the ''p''-power summable sequences, with the ''p''-norm. These are special cases of [[Lp space|L<sup>''p''</sup> spaces]] for the [[counting measure]] on the set of natural numbers. Other important classes of sequences like [[convergent sequence]]s or [[null sequence]]s form sequence spaces, respectively denoted ''c'' and ''c''<sub>0</sub>, 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'''<sup>'''N'''</sup> the set of all sequences of scalars
| |
| :<math>(x_n)_{n\in\mathbf{N}},\quad x_n\in\mathbf{K}.</math>
| |
| This can be turned into a [[vector space]] by defining [[vector addition]] as
| |
| :<math>(x_n)_{n\in\mathbf{N}} + (y_n)_{n\in\mathbf{N}} \stackrel{\rm{def}}{=} (x_n + y_n)_{n\in\mathbf{N}}</math>
| |
| and the [[scalar multiplication]] as
| |
| :<math>\alpha(x_n)_{n\in\mathbf{N}} := (\alpha x_n)_{n\in\mathbf{N}}.</math>
| |
| A '''sequence space''' is any linear subspace of '''K'''<sup>'''N'''</sup>.
| |
| | |
| === ℓ<sup>''p''</sup> spaces===
| |
| For 0 < ''p'' < ∞, ℓ<sup>''p''</sup> is the subspace of '''K'''<sup>'''N'''</sup> consisting of all sequences ''x'' = ('''x'''<sub>''n''</sub>) satisfying
| |
| :<math>\sum_n |x_n|^p < \infty.</math>
| |
| If ''p'' ≥ 1, then the real-valued operation <math>\|\cdot\|_p</math> defined by
| |
| :<math>\|x\|_p = \left(\sum_n|x_n|^p\right)^{1/p}</math>
| |
| defines a [[norm (mathematics)|norm]] on ℓ<sup>''p''</sup>. In fact, ℓ<sup>''p''</sup> is a [[complete metric space]] with respect to this norm, and therefore is a [[Banach space]].
| |
| | |
| If 0 < ''p'' < 1, then ℓ<sup>''p''</sup> does not carry a norm, but rather a [[metric space|metric]] defined by
| |
| :<math>d(x,y) = \sum_n |x_n-y_n|^p.\,</math>
| |
| | |
| If ''p'' = ∞, then ℓ<sup>∞</sup> is defined to be the space of all bounded sequences. With respect to the norm
| |
| :<math>\|x\|_\infty = \sup_n |x_n|,</math>
| |
| ℓ<sup>∞</sup> is also a Banach space.
| |
| | |
| ===''c'' and ''c''<sub>0</sub>===
| |
| The space of [[Limit of a sequence|convergent sequence]]s ''c'' is a sequence space. This consists of all ''x'' ∈ '''K'''<sup>'''N'''</sup> such that lim<sub>''n''→∞</sub>''x''<sub>''n''</sub> exists. Since every convergent sequence is bounded, ''c'' is a linear subspace of ℓ<sup>∞</sup>. 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 ''c''<sub>0</sub> 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 (mathematics)|series]], denote by ''bs'', is the space of sequences ''x'' for which
| |
| :<math>\sup_n \left\vert \sum_{i=0}^n x_i \right\vert < \infty.</math>
| |
| This space, when equipped with the norm
| |
| :<math>\|x\|_{bs} = \sup_n \left\vert \sum_{i=0}^n x_i \right\vert,</math>
| |
| is a Banach space isometrically isomorphic to ℓ<sup>∞</sup>, via the linear mapping
| |
| :<math>(x_n)_{n\in\mathbf{N}} \mapsto \left(\sum_{i=0}^n x_i\right)_{n\in\mathbf{N}}.</math>
| |
| 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 <math>c_{00}</math> 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 set|dense]] in many sequence spaces.
| |
| | |
| == Properties of ℓ<sup>''p''</sup> spaces and the space ''c''<sub>0</sub> ==
| |
| {{see also|c space}}
| |
| The space ℓ<sup>2</sup> is the only ℓ<sup>''p''</sup> space that is a [[Hilbert space]], since any norm that is induced by an inner product should satisfy the parallelogram identity <math>\|x+y\|_p^2 + \|x-y\|_p^2= 2\|x\|_p^2 + 2\|y\|_p^2</math>. Substituting two distinct unit vectors in for ''x'' and ''y'' directly shows that the identity is not true unless ''p'' = 2.
| |
| | |
| Each ℓ<sup>''p''</sup> is distinct, in that ℓ<sup>''p''</sup> is a strict subset of ℓ<sup>''s''</sup> whenever ''p'' < ''s''; furthermore, ℓ<sup>''p''</sup> is not linearly [[isomorphic]] to ℓ<sup>''s''</sup> when ''p'' ≠ ''s''. In fact, by Pitt's theorem {{harv|Pitt|1936}}, every bounded linear operator from ℓ<sup>''s''</sup> to ℓ<sup>''p''</sup> is [[Compact operator|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<sup>''s''</sup>, and is thus said to be [[strictly singular]].
| |
| | |
| If 1 < ''p'' < ∞, then the [[dual space|(continuous) dual space]] of ℓ<sup>''p''</sup> is isometrically isomorphic to ℓ<sup>''q''</sup>, where ''q'' is the [[Hölder conjugate]] of ''p'': 1/''p'' + 1/''q'' = 1. The specific isomorphism associates to an element ''x'' of ℓ<sup>''q''</sup> the functional
| |
| :<math>L_x(y) = \sum_n x_ny_n</math> | |
| for ''y'' in ℓ<sup>''p''</sup>. [[Hölder's inequality]] implies that ''L''<sub>''x''</sup> is a bounded linear functional on ℓ<sup>''p''</sup>, and in fact
| |
| :<math>|L_x(y)| \le \|x\|_q\,\|y\|_p</math>
| |
| so that the operator norm satisfies
| |
| :<math>\|L_x\|_{(\ell^p)^*} \stackrel{\rm{def}}{=}\sup_{y\in\ell^p, y\not=0} \frac{|L_x(y)|}{\|y\|_p} \le \|x\|_q.</math>
| |
| In fact, taking ''y'' to be the element of ℓ<sup>''p''</sup> with
| |
| :<math>y_n = \begin{cases}0&\rm{if}\ x_n=0\\
| |
| x_n^{-1}|x_n|^q &\rm{if}\ x_n\not=0
| |
| \end{cases}</math>
| |
| gives ''L''<sub>''x''</sub>(''y'') = ||''x''||<sub>''q''</sup>, so that in fact
| |
| :<math>\|L_x\|_{(\ell^p)^*} = \|x\|_q.</math>
| |
| Conversely, given a bounded linear functional ''L'' on ℓ<sup>''p''</sup>, the sequence defined by ''x''<sub>''n''</sub> = ''L''(''e''<sub>''n''</sub>) lies in ℓ<sup>''q''</sup>. Thus the mapping <math>x\mapsto L_x</math> gives an isometry
| |
| :<math>\kappa_q : \ell^q \to (\ell^p)^*.</math>
| |
| | |
| The map
| |
| :<math>\ell^q\xrightarrow{\kappa_q}(\ell^p)^*\xrightarrow{(\kappa_q^*)^{-1}}</math>
| |
| obtained by composing κ<sub>''p''</sub> with the inverse of its [[Dual space#Transpose of a continuous linear map|transpose]] coincides with the [[Reflexive space#Definitions|canonical injection]] of ℓ<sup>''q''</sup> into its [[double dual]]. As a consequence ℓ<sup>''q''</sup> is a [[reflexive space]]. By [[abuse of notation]], it is typical to identify ℓ<sup>''q''</sup> with the dual of ℓ<sup>''p''</sup>: (ℓ<sup>''p''</sup>)<sup>*</sup> = ℓ<sup>''q''</sup>. Then reflexivity is understood by the sequence of identifications (ℓ<sup>''p''</sup>)<sup>**</sup> = (ℓ<sup>''q''</sup>)<sup>*</sup> = ℓ<sup>''p''</sup>.
| |
| | |
| The space ''c''<sub>0</sub> is defined as the space of all sequences converging to zero, with norm identical to ||''x''||<sub>∞</sub>. It is a closed subspace of ℓ<sup>∞</sup>, hence a Banach space. The [[dual space|dual]] of ''c''<sub>0</sub> is ℓ<sup>1</sup>; the dual of ℓ<sup>1</sup> is ℓ<sup>∞</sup>. For the case of natural numbers index set, the ℓ<sup>''p''</sup> and ''c''<sub>0</sub> are [[separable space|separable]], with the sole exception of ℓ<sup>∞</sup>. The dual of ℓ<sup>∞</sup> is the [[ba space]].
| |
| | |
| The spaces ''c''<sub>0</sub> and ℓ<sup>''p''</sup> (for 1 ≤ ''p'' < ∞) have a canonical unconditional [[Schauder basis]] {''e''<sub>''i''</sub> | ''i'' = 1, 2,…}, where ''e''<sub>''i''</sub> is the sequence which is zero but for a 1 in the ''i''<sup>th</sup> entry.
| |
| | |
| The space ℓ<sup>''1''</sup> has the [[Schur's property|Schur property]]: In ℓ<sup>''1''</sup>, any sequence that is [[weak convergence|weakly convergent]] is also [[strong convergence|strongly convergent]] {{harv|Schur|1921}}. However, since the [[weak topology]] on infinite-dimensional spaces is strictly weaker than the [[strong topology]], there are [[net (mathematics)|nets]] in ℓ<sup>''1''</sup> that are weak convergent but not strong convergent.
| |
| | |
| The ℓ<sup>''p''</sup> spaces can be [[embedding|embedded]] into many [[Banach space]]s. The question of whether every infinite-dimensional Banach space contains an isomorph of some ℓ<sup>''p''</sup> or of ''c''<sub>0</sub>, was answered negatively by [[Boris Tsirelson|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 (linear algebra)|quotient space]] of ℓ<sup>1</sup>, was answered in the affirmative by {{harvtxt|Banach|Mazur|1933}}. That is, for every separable Banach space ''X'', there exists a quotient map <math>Q:\ell^1 \to X</math>, so that ''X'' is isomorphic to <math>\ell^1 / \ker Q</math>. In general, ker ''Q'' is not complemented in ℓ<sup>1</sup>, that is, there does not exist a subspace ''Y'' of ℓ<sup>1</sup> such that <math>\ell^1 = Y \oplus \ker Q</math>. In fact, ℓ<sup>1</sup> has uncountably many uncomplemented subspaces that are not isomorphic to one another (for example, take <math>X=\ell^p</math>; since there are uncountably many such ''X'' 's, and since no ℓ<sup>''p''</sup> 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<sup>''p''</sup> is that it is not [[polynomially reflexive space|polynomially reflexive]].
| |
| | |
| == See also ==
| |
| | |
| *[[Lp space|L<sup>p</sup> space]]
| |
| *[[Tsirelson space]]
| |
| *[[beta-dual space]]
| |
| | |
| ==References==
| |
| *{{citation|first1=S.|last1=Banach|first2=S.|last2=Mazur|title=Zur Theorie der linearen Dimension|journal=Studia Mathematica|volume=4|year=1933|pages=100–112}}.
| |
| * {{citation|last1=Dunford|first1=Nelson|last2=Schwartz|first2=Jacob T.|title=Linear operators, volume I|publisher=Wiley-Interscience|year=1958}}.
| |
| * {{citation|doi=10.1112/jlms/s1-11.3.174|first=H.R.|last=Pitt|title=A note on bilinear forms|journal=J. London Math. Soc.|volume=11|issue=3|year=1936|pages=174–180}}.
| |
| *{{citation|first=J.|last=Schur|title=Über lineare Transformationen in der Theorie der unendlichen Reihen|journal=Journal für die reine und angewandte Mathematik|volume=151|year=1921|pages=79–111}}.
| |
| | |
| [[Category:Functional analysis]]
| |
| [[Category:Sequences and series]]
| |
| [[Category:Sequence spaces]]
| |