Reflexive Banach Spaces — Mathematics Knowledge System

Overview

A Banach space X is reflexive if the canonical embedding J: X -> X** into its bidual is surjective (hence an isometric isomorphism). Reflexivity is a structural property capturing 'roundness' of the space: it is equivalent to the closed unit ball being weakly compact, which is the key tool for existence results in optimization and PDE. The L^p spaces for 1 < p < infinity are reflexive; L^1 and L^inf are not.

Intuition

The dual X* of a Banach space consists of all bounded linear functionals. The bidual X** is the dual of X*. Every element x in X defines a functional J(x) on X* by evaluation: J(x)(f) = f(x). Reflexivity says every functional on X* arises this way — no 'phantom' elements live in X** beyond those coming from X. Geometrically, reflexivity is tied to the unit ball having no flat sides extending to infinity in the weak topology.

Formal Definition

Definition

The canonical embedding J: X -> X** is defined by J(x)(f) = f(x) for f in X*. It is always an isometric linear injection. X is reflexive if J is surjective, i.e., J(X) = X**. Equivalently, X is reflexive iff the closed unit ball B_X is compact in the weak topology sigma(X, X*).

J: X \to X^{**},\quad J(x)(f) = f(x),\quad \|J(x)\|_{X^{**}} = \|x\|_X

Canonical embedding

X \text{ reflexive} \iff J(X) = X^{**}

Reflexivity condition

\overline{B_X} = \{x \in X : \|x\| \le 1\} \text{ is weakly compact} \iff X \text{ reflexive}

Weak compactness characterisation (Eberlein–Smulian)

Notation

Notation	Meaning
$X^*$	Continuous dual space of X
$X^{**}$	Bidual (double dual) of X
$J$	Canonical embedding X -> X**
$\sigma(X,X^*)$	Weak topology on X

Theorems

Theorem 1.1: Theorem 1.1 (Eberlein–Smulian)

A Banach space X is reflexive if and only if its closed unit ball is weakly compact. Equivalently, every bounded sequence in X has a weakly convergent subsequence.

Theorem 1.2: Theorem 1.2

If X is reflexive, then every closed convex subset of X that is bounded is weakly compact. In particular, every bounded sequence has a weakly convergent subsequence.

Theorem 1.3: Theorem 1.3

L^p(mu) is reflexive for 1 < p < infinity with (L^p)* isometrically isomorphic to L^q where 1/p + 1/q = 1. L^1 and L^inf are not reflexive.

Worked Examples

1
By the Riesz representation theorem, every f in H* has the form f(x) = <x, y> for a unique y in H, and ||f|| = ||y||.
2
Given F in H**, we need x_0 in H with F(f) = f(x_0) for all f in H*.
3
The map g: H* -> R defined by g(f) = F(f) is a bounded linear functional on H*. Identifying H* with H via Riesz, g corresponds to some x_0 in H.
$F(f) = F(\langle \cdot, y \rangle) = \langle x_0, y \rangle = f(x_0)$
4
Hence J(x_0) = F, so J is surjective and H is reflexive.

✓ Answer

Every Hilbert space is reflexive because the Riesz representation theorem gives a surjective isometry H -> H*, making J: H -> H** surjective.

Practice Problems

Mediumproof writing

Show that if X is reflexive and Y is a closed subspace of X, then Y is reflexive.

Mediumfree response

Let X be a reflexive Banach space and f: X -> R a continuous convex function bounded below. Show f attains its infimum on every non-empty closed convex bounded subset C of X.

Quiz

Summary

A Banach space X is reflexive if the canonical embedding J: X -> X** is surjective.
Reflexivity is equivalent to the unit ball being weakly compact (Eberlein–Smulian theorem).
Reflexive spaces include all Hilbert spaces and L^p spaces for 1 < p < infinity.
Non-reflexive spaces include L^1, L^inf, and c_0.
Reflexivity guarantees existence of minimizers for continuous convex functions on closed bounded convex sets.

References

BookBrezis, H. — Functional Analysis, Sobolev Spaces and Partial Differential Equations, Chapter 3
BookConway, J. — A Course in Functional Analysis (2nd ed.), Chapter V
Websiteen.wikipedia.org