Uniform distribution (discrete): Difference between revisions

Revision as of 12:51, 8 January 2014

28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. 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 sequences 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 N}, x_{n} \in K .

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

(x_{n})_{n \in N} + (y_{n})_{n \in N} \overset{d e f}{=} (x_{n} + y_{n})_{n \in N}

and the scalar multiplication as

α (x_{n})_{n \in N} : = (α x_{n})_{n \in 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} = {(\sum_{n} | x_{n} |^{p})}^{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} | \sum_{i = 0}^{n} x_{i} | < \infty .

This space, when equipped with the norm

‖ x ‖_{b s} = \sup_{n} | \sum_{i = 0}^{n} x_{i} |,

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

(x_{n})_{n \in N} \mapsto {(\sum_{i = 0}^{n} x_{i})}_{n \in 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

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In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value. 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 in 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 Template:Harv, 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} ‖_{(ℓ^{p})^{*}} \overset{d e f}{=} \sup_{y \in ℓ^{p}, y = 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 & i f x_{n} = 0 \\ x_{n}^{- 1} | x_{n} |^{q} & i f x_{n} = 0 \end{cases}

gives L_x(y) = ||x||_{q, so that in fact}

‖ L_{x} ‖_{(ℓ^{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

κ_{q} : ℓ^{q} \to (ℓ^{p})^{*} .

The map

ℓ^{q} \overset{κ_{q}}{\to} (ℓ^{p})^{*} \overset{(κ_{q}^{*})^{- 1}}{\to}

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 Template:Harv. 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 Template:Harvtxt. That is, for every separable Banach space X, there exists a quotient map $Q : ℓ^{1} \to X$ , so that X is isomorphic to $ℓ^{1} / \ker Q$ . In general, ker Q is not complemented in ℓ¹, that is, there does not exist a subspace Y of ℓ¹ such that $ℓ^{1} = Y \oplus \ker Q$ . In fact, ℓ¹ has uncountably many uncomplemented subspaces that are not isomorphic to one another (for example, take $X = ℓ^{p}$ ; 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.

References

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15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

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Uniform distribution (discrete): Difference between revisions

Revision as of 12:51, 8 January 2014

Contents

Definition

ℓ^p spaces

c and c₀

Other sequence spaces

Properties of ℓ^p spaces and the space c₀

See also

References

Navigation menu

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+{{for|usage in evolutionary biology|Sequence space (evolution)}}
+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]].
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+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]].
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+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>
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+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>
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+and the [[scalar multiplication]] as
+:<math>\alpha(x_n)_{n\in\mathbf{N}} := (\alpha x_n)_{n\in\mathbf{N}}.</math>
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+A '''sequence space''' is any linear subspace of '''K'''<sup>'''N'''</sup>.
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- </ul>
-== Nike Free 5.0 Nz . ==
+=== 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]].
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+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>
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-== Beats Pill Nz 000. And thirdly ==
+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.
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+===''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''&nbsp;∈&nbsp;'''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.
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-== Nike Roshe Run  WordPress ==
+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.
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+===Other sequence spaces===
+The space of bounded [[series (mathematics)|series]], denote by ''bs'', is the space of sequences ''x'' for which
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+:<math>\sup_n \left\vert \sum_{i=0}^n x_i \right\vert < \infty.</math>
+This space, when equipped with the norm
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+:<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
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+:<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.
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-== Louis Vuitton Wallet Carex pallescens ==
+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.
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+== Properties of ℓ<sup>''p''</sup> spaces and the space ''c''<sub>0</sub> ==
+{{see also|c space}}
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+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''&nbsp;=&nbsp;2.
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+Each ℓ<sup>''p''</sup> is distinct, in that ℓ<sup>''p''</sup> is a strict subset of ℓ<sup>''s''</sup> whenever ''p''&nbsp;<&nbsp;''s''; furthermore, ℓ<sup>''p''</sup> is not linearly [[isomorphic]] to ℓ<sup>''s''</sup> when&nbsp;''p''&nbsp;≠&nbsp;''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''&nbsp;<&nbsp;''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]].
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+If 1&nbsp;< ''p''&nbsp;< ∞, 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''&nbsp;+ 1/''q''&nbsp;= 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>
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+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>
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+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>
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+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\\
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+x_n^{-1}|x_n|^q &\rm{if}\ x_n\not=0
+\end{cases}</math>
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+gives ''L''<sub>''x''</sub>(''y'') = ||''x''||<sub>''q''</sup>, so that in fact
+:<math>\|L_x\|_{(\ell^p)^*} = \|x\|_q.</math>
- </ul>
+Conversely, given a bounded linear functional ''L'' on ℓ<sup>''p''</sup>, the sequence defined by ''x''<sub>''n''</sub>&nbsp;= ''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>
-== Cheap Nike Shoes Singapore 1973 ==
+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>&nbsp;= ℓ<sup>''q''</sup>.  Then reflexivity is understood by the sequence of identifications (ℓ<sup>''p''</sup>)<sup>**</sup>&nbsp;= (ℓ<sup>''q''</sup>)<sup>*</sup>&nbsp;= ℓ<sup>''p''</sup>.
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+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]].
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-== Longchamp Bags  such as the opening prayer ==
+The spaces ''c''<sub>0</sub> and ℓ<sup>''p''</sup> (for 1&nbsp;≤ ''p''&nbsp;< ∞) have a canonical unconditional [[Schauder basis]] {''e''<sub>''i''</sub>&nbsp;| ''i''&nbsp;= 1, 2,…}, where ''e''<sub>''i''</sub> is the sequence which is zero but for a 1 in the ''i''<sup>th</sup> entry.
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+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.
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+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).
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+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]].
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+*[[Lp space|L<sup>p</sup> space]]
+*[[Tsirelson space]]
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+*[[beta-dual space]]
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-== Mbt Australia Also ==
+==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&ndash;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}}.
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Uniform distribution (discrete): Difference between revisions

Revision as of 12:51, 8 January 2014

Definition

ℓp spaces

c and c0

Other sequence spaces

Properties of ℓp spaces and the space c0

See also

References

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Search

ℓ^p spaces

c and c₀

Properties of ℓ^p spaces and the space c₀