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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 M_n of degree n (that is, M_n is n-linear), we can define a polynomial p as

p (x) = M_{n} (x, \dots, x)

(that is, applying M_n 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_n as consisting of all n-homogeneous polynomials.

The P₁ 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 functionals g and h. Therefore, assuming that the scalars are complex numbers, every sequence x_n satisfying g(x_n) → 0 for all linear functionals g, satisfies also f(x_n) → 0 for all quadratic forms f.

In infinite dimension the situation is different. For example, in a Hilbert space, an orthonormal sequence x_n satisfies g(x_n) → 0 for all linear functionals g, and nevertheless f(x_n) = 1 where f is the quadratic form f(x) = ||x||². In more technical words, this quadratic form fails to be weakly sequentially continuous at the origin.

On a reflexive Banach space with the approximation property the following two conditions are equivalent:^[1]

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 $ℓ^{p}$ spaces, the P_n is reflexive if and only if Template:Mvar < Template:Mvar. Thus, no $ℓ^{p}$ is polynomially reflexive. ( $ℓ^{\infty}$ is ruled out because it is not reflexive.)

Thus if a Banach space admits $ℓ^{p}$ as a quotient space, it is not polynomially reflexive. This makes polynomially reflexive spaces rare.

The Tsirelson space T* is polynomially reflexive.^[2]

Notes

↑ Farmer 1994, page 261.
↑ Alencar, Aron and Dineen 1984.

References

Alencar, R., Aron, R. and S. Dineen (1984), "A reflexive space of holomorphic functions in infinitely many variables", Proc. Amer. Math. Soc. 90: 407–411.

Farmer, Jeff D. (1994), "Polynomial reﬂexivity in Banach spaces", Israel Journal of Mathematics 87: 257–273. Template:MathSciNet

Jaramillo, J. and Moraes, L. (2000), "Dualily and reflexivity in spaces of polynomials", Arch. Math. (Basel) 74: 282–293. Template:MathSciNet

Mujica, Jorge (2001), "Reflexive spaces of homogeneous polynomials", Bull. Polish Acad. Sci. Math. 49:3, 211–222. Template:MathSciNet

[1] Farmer 1994, page 261.

[2] Alencar, Aron and Dineen 1984.

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[2]

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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]]