Reflexivity means that passing to the double dual loses no points. It is the condition under which weak compactness of bounded sets begins to resemble finite-dimensional compactness.
Canonical embedding
Every Banach space maps into its double dual by evaluation. Reflexivity asks whether this canonical embedding is onto.
Definition. A Banach space \(X\) is reflexive if the canonical isometry \(J_X:X\to X^{**}\), \(J_Xx(f)=f(x)\), is onto. Equivalently, every continuous linear functional on \(X^*\) is evaluation at a unique point of \(X\).
Proposition. Finite-dimensional normed spaces are reflexive.
Proof. If \(\dim X=n\), then \(\dim X^*=n\) and \(\dim X^{**}=n\). The canonical map \(J_X\) is injective by Hahn-Banach and is a linear map between vector spaces of the same finite dimension, hence onto.
Goldstine theorem
Goldstine's theorem says that the original space is always weak-star dense in the double dual unit ball. Reflexivity strengthens density into equality.
Theorem. The image \(J_X(B_X)\) is weak-star dense in \(B_{X^{**}}\).
Proof. It suffices to meet a basic weak-star neighborhood of \(F\in B_{X^{**}}\), determined by \(f_1,\ldots,f_n\in X^*\) and \(\varepsilon>0\). Consider \(S:X\to\mathbb F^n\), \(Sx=(f_1(x),\ldots,f_n(x))\). If the vector \((F(f_1),\ldots,F(f_n))\) were not in the closure of \(S((1+\varepsilon)B_X)\), finite-dimensional separation would give scalars \(a_i\) such that the functional \(g=\sum_i a_if_i\) satisfies \(|F(g)|>(1+\varepsilon)\|g\|\), contradicting \(\|F\|\le1\). Hence there is \(x\) with \(\|x\|\le1+\varepsilon\) and \(|f_i(x)-F(f_i)|<\varepsilon\). Scaling \(x\) by \(1+\varepsilon\) and absorbing the resulting small error gives a point of \(B_X\) in the prescribed neighborhood.
Weak compactness criterion
Weak compactness of the unit ball is the geometric signature of reflexivity. This is the infinite-dimensional replacement for compactness of bounded closed sets.
Theorem. A Banach space \(X\) is reflexive iff its closed unit ball is compact in the weak topology.
Proof. If \(X\) is reflexive, then \(J_X(B_X)=B_{X^{**}}\), and the weak topology on \(B_X\) corresponds to the weak-star topology on \(B_{X^{**}}\). Banach-Alaoglu makes \(B_{X^{**}}\) weak-star compact, so \(B_X\) is weakly compact.
Conversely, suppose \(B_X\) is weakly compact. The map \(J_X:(B_X,\sigma(X,X^*))\to(B_{X^{**}},\sigma(X^{**},X^*))\) is continuous, so \(J_X(B_X)\) is weak-star compact and therefore weak-star closed. By Goldstine theorem it is weak-star dense in \(B_{X^{**}}\), hence \(J_X(B_X)=B_{X^{**}}\). Scaling gives \(J_X(X)=X^{**}\), so \(X\) is reflexive.
Closed subspaces
Closed subspaces preserve the ambient analytic structure. They are the subspaces for which limits, quotients, and orthogonal or weak arguments behave correctly.
Proposition. Every closed subspace \(M\) of a reflexive Banach space \(X\) is reflexive.
Proof. The closed unit ball \(B_M\) is weakly closed in \(X\), because closed convex sets are weakly closed. It is contained in \(B_X\), which is weakly compact. Therefore \(B_M\) is weakly compact. The weak topology inherited from \(X\) agrees with \(\sigma(M,M^*)\), because every \(m^*\in M^*\) extends to some \(x^*\in X^*\) by Hahn-Banach. The weak compactness criterion gives reflexivity of \(M\).
Quotients
Reflexivity also passes to quotients. The annihilator relations make the quotient double dual visible inside the original double dual.
Proposition. If \(X\) is reflexive and \(M\subset X\) is closed, then \(X/M\) is reflexive.
Proof. The quotient map \(q:X\to X/M\) is weak-to-weak continuous because every functional on \(X/M\) pulls back to a functional on \(X\). Hence \(q(rB_X)\) is weakly compact for every \(r>0\). The closed unit ball of \(X/M\) equals \(\bigcap_{n\ge1}q((1+1/n)B_X)\): inclusion \(\subset\) follows from the definition of quotient norm by choosing representatives of norm at most \(1+1/n\), and the reverse inclusion is immediate. This intersection is weakly closed inside the weakly compact set \(q(2B_X)\), hence weakly compact. The weak compactness criterion gives reflexivity.
Uniform convexity
Uniform convexity is a quantitative curvature condition. It forces reflexivity by ruling out flat behavior in bounded sequences.
Theorem. Every uniformly convex Banach space is reflexive.
Proof. This is the Milman-Pettis theorem. Let \((C_i)\) be a decreasing family of nonempty closed bounded convex sets. Uniform convexity implies normal structure: a closed bounded convex set with more than one point contains a point whose maximal distance to the set is strictly smaller than the diameter. This prevents the finite-intersection construction from losing all points and implies weak compactness of closed balls. The weak compactness criterion then yields reflexivity. In \(L^p\) and \(\ell^p\), \(1<p<\infty\), uniform convexity follows from Clarkson inequalities, so these spaces are reflexive.
Nonreflexive spaces
Nonreflexive examples show what can go wrong: bounded sets may fail to be weakly compact, and the double dual contains genuinely new elements.
Example. The spaces \(\ell^1\), \(c_0\), and \(L^1[0,1]\) are not reflexive. For instance, \(c_0^*=\ell^1\) and \((c_0)^{**}\cong\ell^\infty\), so the canonical embedding misses bounded sequences that do not converge to \(0\). In \(\ell^1\), weak compactness of the unit ball would force a weakly convergent subnet of the unit vectors, but mass can escape to infinity.