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Polynomially reflexive space

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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.

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Given a multilinear functional Mn of degree n (that is, Mn is n-linear), we can define a polynomial p as

p ( x ) = M n ( x , … , x )

(that is, applying Mn 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 Pn 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 xn satisfying g(xn) → 0 for all linear functionals g, satisfies also f(xn) → 0 for all quadratic forms f.

In infinite dimension the situation is different. For example, in a Hilbert space, an orthonormal sequence xn satisfies g(xn) → 0 for all linear functionals g, and nevertheless f(xn) = 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 ℓ p spaces, the Pn is reflexive if and only if n < p. Thus, no ℓ p is polynomially reflexive. ( ℓ ∞ 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.

    References

    Polynomially reflexive space Wikipedia
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