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 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 ''P''₁ 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:
* 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 $\ell^p$ spaces, the ''P''_''n'' is reflexive if and only if < . Thus, no $\ell^p$ is polynomially reflexive. ($\ell^\infty$ is ruled out because it is not reflexive.)

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

The Tsirelson space ''T''* is polynomially reflexive.Alencar, Aron and Dineen 1984.

Notes

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.
*Jaramillo, J. and Moraes, L. (2000), "Dualily and reflexivity in spaces of polynomials", ''Arch. Math. (Basel)'' 74: 282–293.
*Mujica, Jorge (2001), "Reflexive spaces of homogeneous polynomials", ''Bull. Polish Acad. Sci. Math.'' 49:3, 211–222.

{{Functional analysis

Banach spaces