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In [[mathematics]], a '''polynomially reflexive space''' is a [[Banach space]] ''X'', on which the space of all polynomials in each degree is a [[reflexive space]].
 
Given a [[multilinear]] [[functional (mathematics)|functional]] ''M''<sub>''n''</sub> of degree ''n'' (that is, ''M''<sub>''n''</sub> is ''n''-linear), we can define a polynomial ''p'' as
 
:<math>p(x)=M_n(x,\dots,x)</math>
 
(that is, applying ''M''<sub>''n''</sub> on the ''[[diagonal]]'') or any finite sum of these. If only ''n''-linear functionals are in the sum, the polynomial is said to be ''n''-homogeneous.
 
We define the space ''P''<sub>''n''</sub> as consisting of all ''n''-homogeneous polynomials.
 
The ''P''<sub>1</sub> is identical to the [[dual space]], and is thus reflexive for all reflexive ''X''. This implies that reflexivity is a prerequisite for polynomial reflexivity.
 
==Relation to continuity of forms==
 
On a finite-dimensional linear space, a [[quadratic form]] ''x''↦''f''(''x'') is always a (finite) linear combination of products ''x''↦''g''(''x'') ''h''(''x'') of two [[linear functional]]s ''g'' and ''h''. Therefore, assuming that the scalars are complex numbers, every sequence ''x<sub>n</sub>'' satisfying ''g''(''x<sub>n</sub>'') &rarr; 0 for all linear functionals ''g'', satisfies also ''f''(''x<sub>n</sub>'') &rarr; 0 for all quadratic forms ''f''.
 
In infinite dimension the situation is different. For example, in a [[Hilbert space]], an [[orthonormal]] sequence ''x<sub>n</sub>'' [[Weak convergence (Hilbert space)#Weak convergence of orthonormal sequences|satisfies]] ''g''(''x<sub>n</sub>'') &rarr; 0 for all linear functionals ''g'', and nevertheless ''f''(''x<sub>n</sub>'') = 1 where ''f'' is the quadratic form ''f''(''x'') = ||''x''||<sup>2</sup>. In more technical words, this quadratic form fails to be [[Weak convergence (Hilbert space) |weakly]] [[Continuous function (topology)#Sequences and nets|sequentially continuous]] at the origin.
 
On a [[Reflexive space|reflexive]] [[Banach space]] with the [[approximation property]] the following two conditions are equivalent:<ref>Farmer 1994, page 261.</ref>
* every quadratic form is weakly sequentially continuous at the origin;
* the Banach space of all quadratic forms is reflexive.
 
Quadratic forms are 2-homogeneous polynomials. The equivalence mentioned above holds also for ''n''-homogeneous polynomials, ''n''=3,4,...
 
== Examples ==
 
For the <math>\ell^p</math> [[lp space|spaces]], the ''P''<sub>''n''</sub> is reflexive if and only if {{mvar|n}} < {{mvar|p}}. Thus, no <math>\ell^p</math> is polynomially reflexive. (<math>\ell^\infty</math> is ruled out because it is not reflexive.)
 
Thus if a Banach space admits <math>\ell^p</math> as a [[quotient space (linear algebra)|quotient space]], it is not polynomially reflexive. This makes polynomially reflexive spaces rare.
 
The [[Tsirelson space]] ''T''* is polynomially reflexive.<ref>Alencar, Aron and Dineen 1984.</ref>
 
==Notes==
<references />
 
== References ==
 
*Alencar, R., Aron, R. and S. Dineen (1984), "A reflexive space of holomorphic functions in infinitely many variables", ''Proc. Amer. Math. Soc.'' '''90''': 407&ndash;411.
 
*Farmer, Jeff D. (1994), "Polynomial reflexivity in Banach spaces", ''Israel Journal of Mathematics'' '''87''': 257&ndash;273. {{MathSciNet|id=1286830}}
 
*Jaramillo, J. and Moraes, L. (2000), "Dualily and reflexivity in spaces of polynomials", ''Arch. Math. (Basel)'' '''74''': 282&ndash;293. {{MathSciNet|id=1742640}}
 
*Mujica, Jorge (2001), "Reflexive spaces of homogeneous polynomials", ''Bull. Polish Acad. Sci. Math.'' '''49''':3, 211&ndash;222. {{MathSciNet|id=1863260}}
 
[[Category:Banach spaces]]

Revision as of 15:19, 17 February 2014

Making a computer run rapidly is very easy. Most computers run slow considering they are jammed up with junk files, which Windows has to look through each time it wants to locate something. Imagine having to find a book inside a library, but all library books are in a big big pile. That's what it's like for your computer to find anything, when your program is full of junk files.

Many of the reliable companies may provide a full money back guarantee. This means that we have the chance to get your money back should you find the registry cleaning has not delivered what we expected.

H/w associated error handling - when hardware causes BSOD installing newest fixes for the hardware and/ or motherboard could enable. You may moreover add new hardware that is compatible with the system.

Paid registry cleaners found on the alternative hand, I have found, are often cheap. They offer normal, free changes or at least cheap updates. This follows because the software maker must confirm their product is many effective in staying ahead of its competitors.

In a word, to accelerate windows XP, Vista business, it's very significant to disable several business products and clean and optimize the registry. You are able to follow the procedures above to disable unnecessary programs. To optimize the registry, I suggest you employ a tuneup utilities software. Because it is very rather dangerous for you to edit the registry by oneself.

Another key element whenever you compare registry cleaners is having a facility to manage a start-up tasks. This just means which you can select what programs you'd like to commence whenever you begin the PC. If you have unnecessary programs starting when you boot up your PC this can lead to a slow running computer.

Google Chrome is my lifeline and to the day fortunately. My all settings plus research connected bookmarks were saved inside Chrome plus stupidly I did not synchronize them with all the Gmail to shop them online. I could not afford to install modern adaptation and sacrifice all my function settings. There was no way to retrieve the aged settings. The only option left for me was to miraculously fix it browser inside a method that all data plus settings stored inside it are recovered.

Before you purchase a entire hot system; it is very time to get the older 1 cleaned up to start getting more done online today! Visit our site under plus access the many reputable registry cleaner software available.

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