Reflexive Banach space - Imported Knowlpedia Content

Definition

Let $X$ be a Banach space over $\mathbb R$ or $\mathbb C$ , let $X'$ be its continuous dual, and let $X''=(X')'$ . The canonical evaluation map

J_X:X\to X'',\qquad (J_Xx)(\varphi)=\varphi(x),

is a linear isometry. The space $X$ is reflexive if $J_X$ is surjective. Equivalently, every continuous linear functional on $X'$ is evaluation at a unique vector of $X$ . Reflexivity is therefore a property of the Banach-space topology, not merely an algebraic identification with a double dual.

Equivalent characterizations

A Banach space is reflexive exactly when its closed unit ball is compact for the weak topology. By the Eberlein–Šmulian theorem, this is also equivalent to weak sequential compactness of the unit ball. These are theorem-level characterizations, not parts of the definition Conway, Chapter V.

Examples and permanence

Every finite-dimensional Banach space and every Hilbert space is reflexive. For a measure space, $L^p$ is reflexive when $1<p<\infty$ ; in contrast, $\ell^1$ , $c_0$ , and infinite-dimensional $L^1$ spaces are standard nonexamples. Closed subspaces and Banach-space quotients of a reflexive space are reflexive. Moreover, $X$ is reflexive exactly when $X'$ is reflexive Rudin, Chapter 4.

Conventions and scope

Some authors call a locally convex space reflexive when a canonical map into a suitably topologized bidual is an isomorphism. That broader notion depends on which dual topology is chosen. This knowl uses the Banach-space convention, where both duals carry their norm topologies and surjectivity of $J_X$ is the defining condition.

References

John B. Conway, A Course in Functional Analysis, 2nd ed., Graduate Texts in Mathematics 96, Springer, 1990. Springer DOI record. Relevant: Chapter V, “Weak Topologies.”
Walter Rudin, Functional Analysis, 2nd ed., McGraw–Hill, 1991. WorldCat record. Relevant: Chapter 4, reflexivity and weak compactness.