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Polynomially Reflexive SpaceIn mathematics, a polynomially reflexive space is a Banach space X, on which all polynomials are reflexive. Given a multilinear functional Mn of degree n (that is, Mn is n-linear), we can define a polynomial p as -
(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. In the presence of the approximation property of X, a reflexive Banach space is polynomially reflexive, if and only if every polynomial on X is weak sequentially continuous. Examples For the spaces, the Pn is reflexive if and only if n < p. Thus, no is polynomially reflexive. ( is ruled out because it is not reflexive.) Thus if space contains as a quotient space, it is not polynomially reflexive. This makes polynomially reflexive spaces rare. The symmetric Tsirelson space is polynomially reflexive.
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