Quasireflexivity

Quasireflexivity is a concept in functional analysis, specifically within the study of Banach spaces. A Banach space X is said to be quasireflexive if its bidual X is reflexive. The bidual X of a Banach space X is the dual space of its dual space X, and it is also a Banach space. The bidual X is related to X via the canonical embedding map J: X -> X, which is an isometric linear embedding.

A space is reflexive if the canonical embedding J is surjective. If X is reflexive, then X is linearly isometric to X, and thus X is also reflexive. However, the converse is not true. A quasireflexive space is one where X is reflexive, but X itself may not be. This means that the bidual space is "better behaved" in terms of reflexivity than the original space.

The codimension of the image of the canonical embedding J in X is finite for quasireflexive spaces. Specifically, if X is quasireflexive, then dim(X / J(X)) is finite. This finite-dimensional difference is a key characteristic of quasireflexive spaces.

An example of a quasireflexive space that is not reflexive is the space of bounded sequences c. Its bidual c is the space of convergent sequences, which is reflexive. The difference between c and J(c) is one-dimensional. Another important example is the space of convergent series l1. Its bidual l1 is reflexive, and the codimension of J(l1) in l1 is 1.

The concept of quasireflexivity provides a way to categorize Banach spaces based on the reflexivity properties of their biduals, highlighting a relationship between a space and its higher duals.