Quasireflexive

Quasireflexive, in mathematics, is most commonly encountered as quasi-reflexivity in the theory of Banach spaces. It describes a relaxation of reflexivity for a Banach space, quantified by how far the natural embedding of the space into its double dual fails to be surjective.

In precise terms, a Banach space X is quasi-reflexive if the canonical embedding J: X → X has finite codimension, equivalently the quotient space X / J(X) is finite dimensional. The dimension of this quotient is called the defect or order of quasi-reflexivity. If the quotient is zero, X is reflexive; if it is finite and nonzero, X is quasi-reflexive of that finite order. Finite-dimensional spaces are quasi-reflexive with defect 0, while non-reflexive spaces may be quasi-reflexive with defect greater than 0.

The concept is central because quasi-reflexive spaces retain many useful structural properties of reflexive spaces while allowing a controlled degree of non-reflexivity. Classic examples include the James space, which is quasi-reflexive of order 1, and other constructed spaces that exhibit higher-order quasi-reflexivity. Quasi-reflexive spaces often arise in the study of duality, approximation properties, and the geometry of Banach spaces, where the behavior of X relative to X provides insight into compactness, weak topologies, and decompositions.

Beyond Banach spaces, the term quasi-reflexive has appeared in other mathematical contexts with different definitions, but such uses are less standardized. In functional analysis, the quasi-reflexive notion remains the most developed and widely cited.

In summary, quasireflexive (or quasi-reflexive) spaces generalize reflexive spaces by allowing a finite-dimensional gap between a space and its bidual, offering a framework to explore near-reflexive phenomena with precise defect measures.