# 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

(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*) → 0 for all linear functionals

_{n}*g*, satisfies also

*f*(

*x*) → 0 for all quadratic forms

_{n}*f*.

In infinite dimension the situation is different. For example, in a Hilbert space, an orthonormal sequence *x _{n}* satisfies

*g*(

*x*) → 0 for all linear functionals

_{n}*g*, and nevertheless

*f*(

*x*) = 1 where

_{n}*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
spaces, the *P*_{n} is reflexive if and only if n < p. Thus, no
is polynomially reflexive. (
is ruled out because it is not reflexive.)

Thus if a Banach space admits 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. MR1286830 - Jaramillo, J. and Moraes, L. (2000), "Dualily and reflexivity in spaces of polynomials",
*Arch. Math. (Basel)***74**: 282–293. MR1742640 - Mujica, Jorge (2001), "Reflexive spaces of homogeneous polynomials",
*Bull. Polish Acad. Sci. Math.***49**:3, 211–222. MR1863260