Editing Polynomially reflexive space

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 map|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}}

{{Functional analysis}}

[[Category:Banach spaces]]