Fact-checked by Grok 2 weeks ago

Reflexive space

In functional analysis, a reflexive space is a Banach space X for which the canonical evaluation map J: X \to X^{**}, defined by J(x)(f) = f(x) for all x \in X and f \in X^* (where X^* is the dual space and X^{**} the bidual), is surjective, thereby establishing an isometric isomorphism between X and its bidual. This property ensures that every continuous linear functional on the dual space X^* arises from evaluation at some element of X. The concept was first introduced by Hans Hahn in 1927 as part of early developments in the theory of normed linear spaces. Reflexive spaces exhibit several significant structural properties that distinguish them from general Banach spaces. Notably, the closed unit ball of a reflexive space is weakly compact, meaning every sequence in the unit ball has a weakly convergent subsequence, which aligns their behavior closely with that of finite-dimensional spaces. Additionally, reflexivity is preserved under taking closed subspaces and duals: if X is reflexive, then so is its dual X^*, and every closed subspace of X is reflexive. All finite-dimensional normed spaces and Hilbert spaces are reflexive, providing foundational examples in the theory. Prominent examples of infinite-dimensional reflexive spaces include the Lebesgue spaces L^p(\mu) for $1 < p < \infty, where reflexivity follows from the uniform convexity of these spaces under the p-norm. In contrast, spaces such as \ell^1 (sequences absolutely summable) and c_0 (sequences converging to zero with the sup norm) are non-reflexive, illustrating that reflexivity is not a universal property of Banach spaces. These characteristics make reflexive spaces particularly useful in applications involving weak convergence, optimization, and operator theory, where the identification with the bidual simplifies many analytical arguments.

Core Concepts

Definition via dual spaces

In functional analysis, the dual space X^* of a normed vector space X over the scalar field \mathbb{K} (typically \mathbb{R} or \mathbb{C}) is the vector space consisting of all continuous linear functionals on X, that is, all bounded linear maps f: X \to \mathbb{K}. Equipped with the operator norm \|f\|_{X^*} = \sup_{\|x\|_X \leq 1} |f(x)|, X^* becomes a Banach space even if X is merely normed. This construction captures the linear structure of X through its "observations" via functionals, forming the foundation for duality theory. The bidual X^{**} is defined as the dual space of X^*, comprising all continuous linear functionals g: X^* \to \mathbb{K}. Thus, elements of X^{**} act on functionals in X^*, extending the duality one level further. For any normed space X, there exists a canonical evaluation map J: X \to X^{**}, known as the natural embedding, given by J(x)(f) = f(x) for all x \in X and f \in X^*. This map is always linear and norm-preserving (an isometry), embedding X isometrically into X^{**} as a subspace (closed if X is complete). Algebraically, J identifies each point in X with the functional on X^* that evaluates it; topologically, since J is an isometry, it preserves the norm topology, ensuring that the image J(X) inherits the metric structure of X. A normed space X is reflexive if the canonical map J: X \to X^{**} is surjective, meaning every continuous linear functional on X^* arises as an evaluation from some element of X. In this case, J is a linear isometry onto its image, and surjectivity implies that X is isometrically isomorphic to X^{**}, aligning the algebraic and topological structures perfectly. For incomplete normed spaces, reflexivity requires this surjectivity, but the property is most studied in complete (Banach) spaces, where X^{**} is automatically Banach and surjectivity yields a Banach isomorphism. The canonical embedding J thus serves as the bridge between X and its iterated duals. The concept of reflexivity originated in the work of Hans Hahn in 1927, who introduced the canonical map and identified spaces where it is surjective (initially termed "regular" spaces). The modern term "reflexive" was coined by Edgar R. Lorch in 1939 to describe this property precisely.

Canonical embedding and reflexivity

In the context of dual spaces, the canonical embedding provides a natural way to identify a normed vector space X with a subspace of its bidual X^{**}. This map, denoted J: X \to X^{**}, is defined by J(x)(\phi) = \phi(x) for all x \in X and \phi \in X^*, where X^* is the continuous dual of X. Thus, J(x) acts as the evaluation functional on X^* at the point x, embedding X linearly into the space of all continuous linear functionals on X^*. A space X is reflexive if and only if this canonical embedding J is surjective, meaning that every continuous linear functional on X^* can be represented as an evaluation at some point in X. In the case of Banach spaces, surjectivity of J implies bijectivity, yielding an isometric isomorphism between X and X^{**}. The map J is always linear and continuous, with \|J(x)\| = \|x\| for all x \in X, making it an isometry. To see the norm preservation, note that \|J(x)\| = \sup_{\|\phi\| \leq 1} |J(x)(\phi)| = \sup_{\|\phi\| \leq 1} |\phi(x)| = \|x\|, where the final equality follows from the dual norm definition. Continuity then holds since J is a linear isometry into the normed space X^{**}. Moreover, J is injective: if J(x) = 0, then \phi(x) = 0 for all \phi \in X^*, so x = 0 by the Hahn-Banach separation theorem in normed spaces. Reflexivity also manifests in topological terms involving the weak* topology on X^{**}, which is the weakest topology making all evaluations \psi \mapsto \psi(\phi) continuous for \phi \in X^*. Under this topology, the image J(B_X) of the closed unit ball B_X = \{x \in X : \|x\| \leq 1\} coincides with the closed unit ball B_{X^{**}} when X is reflexive, ensuring that J(B_X) is weak* closed and compact by the Banach-Alaoglu theorem. In non-reflexive spaces, J(B_X) is proper and only weak* dense in B_{X^{**}}.

Banach Space Reflexivity

Fundamental properties

A fundamental property of reflexive Banach spaces is that their closed unit ball is weakly compact. This characterization, known as , provides an equivalent condition for reflexivity in terms of topological compactness. Reflexivity is preserved under isomorphisms: if two Banach spaces are linearly isomorphic, then one is reflexive if and only if the other is. Additionally, every closed subspace of a reflexive Banach space is itself reflexive, as the canonical embedding restricts appropriately to the subspace. James' distortion theorem asserts that if an infinite-dimensional Banach space contains a isomorphic to \ell^1 (or c_0), then for any \varepsilon > 0, it contains a that is (1 + \varepsilon)-isomorphic to \ell^1 (or c_0). This result highlights structural rigidity in non-reflexive spaces and limits how much can occur without these classical spaces almost isometrically. Reflexive Banach spaces possess the Radon-Nikodym property, which means that every closed bounded subset of the space is dentable and that vector measures taking values in the space admit densities under suitable scalar measures. This property follows from the reflexivity ensuring the existence of supporting hyperplanes in a controlled manner. In a reflexive , the closed unit ball is weakly compact, and by the Eberlein–Šmulian theorem, it is weakly sequentially compact, meaning every sequence in the unit ball has a weakly convergent . However, does not generally imply strong convergence on the unit sphere.

Examples of reflexive Banach spaces

provide a fundamental class of reflexive Banach spaces. The establishes that every Hilbert space H is isometrically isomorphic to its H^*, via the that sends each x \in H to the functional \phi_x(y) = \langle x, y \rangle for y \in H, implying that the canonical into the bidual H^{**} is surjective, hence reflexive. The Lebesgue spaces L^p(\mu) for $1 < p < \infty, where \mu is a \sigma-finite measure, are reflexive Banach spaces. Reflexivity follows from showing that the dual of L^p is L^q with $1/p + 1/q = 1, using to pair elements appropriately, and then verifying the bidual identification. Sobolev spaces W^{k,p}(\Omega) for integers k \geq 1, open sets \Omega \subset \mathbb{R}^n, and $1 < p < \infty are reflexive as closed subspaces of the product space [L^p(\Omega)]^{\binom{n+k}{k}}, which is reflexive. The closed embedding preserves reflexivity since closed subspaces of reflexive spaces are reflexive. All finite-dimensional Banach spaces are reflexive, including spaces like \ell_p^n for $1 \leq p \leq \infty and n < \infty. This holds because the algebraic dimension equals that of the dual and bidual, making the canonical embedding an isomorphism. Uniformly convex Banach spaces form another important class of reflexive spaces, as established by the Milman-Pettis theorem. For example, L^p spaces for $2 < p < \infty are uniformly convex (hence reflexive) via the Clarkson inequalities, while L^2 is both uniformly convex and a Hilbert space.

Non-reflexive Banach spaces

A prominent example of a non-reflexive Banach space is \ell^1, the space of absolutely summable real sequences equipped with the norm \|a\|_1 = \sum |a_n|. The dual of \ell^1 is isometrically isomorphic to \ell^\infty, the space of bounded sequences with the supremum norm, via the pairing \langle a, y \rangle = \sum a_n y_n. The bidual (\ell^1)^{**} is then (\ell^\infty)^*, and the canonical embedding J: \ell^1 \to (\ell^1)^{**} given by Jx(\phi) = \phi(x) for \phi \in (\ell^1)^* = \ell^\infty is an isometric isomorphism onto its image but not surjective. To see the failure of surjectivity, note that there exist bounded linear functionals on \ell^\infty that do not arise from elements of \ell^1; for instance, a Banach limit is a norm-one linear functional \Lambda: \ell^\infty \to \mathbb{R} that extends the ordinary limit on convergent sequences, satisfies \Lambda(y) = \Lambda(Sy) for the shift operator S, and agrees with limits where they exist, but no sequence in \ell^1 can represent it because such representations would vanish on certain ultrafilter-defined measures or fail translation invariance. Similarly, the space c_0 of real sequences converging to zero, normed by the supremum, is non-reflexive. Its dual is isometrically isomorphic to \ell^1 via the same pairing as above, restricted to sequences in c_0, and the bidual is (\ell^1)^* = \ell^\infty. The canonical embedding J: c_0 \to c_0^{**} \cong \ell^\infty maps a sequence x \in c_0 to the functional Jy(x) = \sum y_n x_n for y \in \ell^1, but the image is precisely c_0 itself under the identification, which is a proper subspace of \ell^\infty (e.g., constant sequences like the all-ones sequence lie in \ell^\infty but not in c_0). Thus, J fails to be surjective. The space C[0,1] of continuous real-valued functions on the unit interval with the supremum norm provides another example, as it contains a subspace isomorphic to c_0. Since reflexivity is inherited by isomorphic subspaces (if X is reflexive and Y \cong Z \subset X, then Y is reflexive), the presence of a non-reflexive subspace like c_0 implies C[0,1] cannot be reflexive. A concrete embedding arises from sequences of "tent" functions peaking at distinct dyadic rationals and narrowing to zero height, yielding an isomorphic copy of c_0. Finally, the Lebesgue space L^1[0,1] of integrable functions on [0,1] with the L^1-norm is non-reflexive. Its dual is L^\infty[0,1], and the bidual is (L^\infty[0,1])^*, with the canonical embedding J: L^1[0,1] \to (L^1[0,1])^{**} not surjective. Specifically, there exist elements in the bidual corresponding to finitely additive signed measures on [0,1] that extend the integral representation but are not absolutely continuous with respect to Lebesgue measure, hence not representable by integration against an L^1 function; this failure is tied to L^1[0,1] lacking the Radon-Nikodym property.

Superreflexive Banach spaces

A superreflexive Banach space is defined as a Banach space X such that every Banach space finitely representable in X is reflexive. This notion was introduced by in the early 1970s as a refinement of reflexivity, capturing spaces with stronger geometric properties that prevent the finite representability of non-reflexive structures. An equivalent characterization is that X is superreflexive if and only if there exists a constant K \geq 1 such that for all n \in \mathbb{N}, X contains no K-isomorphic copy of \ell_n^1 or \ell_n^\infty. This condition ensures that as the dimension n increases, subspaces mimicking the extreme cases of \ell^p spaces (with p \to 1 or p \to \infty) cannot embed with uniformly bounded distortion, highlighting the uniform control over non-reflexive embeddings in superreflexive spaces. Superreflexive spaces thus exhibit boundedly complete distortion properties, meaning any potential distortion of non-reflexive finite-dimensional models remains controlled and incompatible with reflexivity preservation. A key property is that every superreflexive Banach space admits an equivalent renorming that makes it uniformly convex. Conversely, any Banach space admitting an equivalent uniformly convex norm is superreflexive. This equivalence underscores the geometric stability of superreflexivity, as uniformly convex norms enforce strict separation of points, aligning with the absence of pathological finite representability. Superreflexivity implies reflexivity, since reflexive spaces form the base case of finite representability preservation, but the converse does not hold. The first counterexample to the converse was constructed by in 1973, yielding a separable reflexive lacking the approximation property; such a space cannot admit an equivalent uniformly convex norm and is thus non-superreflexive. This example demonstrates that reflexivity alone does not guarantee the stronger uniform convexity or finite representability control inherent to superreflexivity.

James' theorem on finite trees

James' theorem on finite trees provides a metric characterization of reflexivity in Banach spaces through the absence of certain finite-dimensional structures with controlled distortion. Specifically, a Banach space X is reflexive if and only if for every \varepsilon > 0, X does not contain a normalized finite \varepsilon- of arbitrary size with branching constant greater than $1 + \varepsilon. This result, established by R. C. James in 1964, links the canonical embedding into the bidual with geometric properties of finite metric embedded in the space. A normalized finite \varepsilon-tree in a is a finitely branching metric consisting of indexed by finite sequences from a , equipped with vectors x_\sigma \in X such that \|x_\sigma\| = 1 for all \sigma, the parent satisfies x_\sigma = \frac{1}{k} \sum_{i=1}^k x_{\sigma^\smallfrown i} where k is the branching degree and \sigma^\smallfrown i are the children, and siblings at each level are separated by \|x_{\sigma^\smallfrown i} - x_{\sigma^\smallfrown j}\| \ge \varepsilon for i \ne j. The branching at a is the supremum of the ratios \frac{\|\sum x_{\sigma^\smallfrown i}\| / k}{\|x_\sigma\|} over branches, measuring the distortion in norm preservation under averaging; a exceeding $1 + \varepsilon indicates significant subadditivity failure in the norm. These structures generalize finite-dimensional approximations of infinite-dimensional phenomena like unconditional convergence. The proof of James' theorem proceeds in two directions. If X is non-reflexive, then by properties of the bidual , X admits a isomorphic to \ell_1, which embeds \ell_1-trees—finite where the is additive (branching constant close to 1)—but with extensions to higher distortion allowing branching constants >1 + \varepsilon for small \varepsilon > 0, violating reflexivity conditions derived from weak compactness. Conversely, if X is reflexive, every is weakly relatively compact, preventing the construction of such distorting finite \varepsilon-trees of unbounded size, as they would imply a non-weakly compact via the tree branches. This relies on James' earlier characterizations linking reflexivity to attainment and weak sequential . This theorem has significant applications in the study of spreading models, where finite tree embeddings help identify asymptotic \ell_p-structures or mixed bases in reflexive spaces, and in unconditional bases, as the absence of high-distortion trees ensures that bases behave well under unconditional convergence without \ell_1-distortions. For example, in Hilbert spaces like \ell_2, no such \varepsilon-trees exist beyond trivial depth due to the parallelogram law enforcing branching constants bounded by 1. In the , Odell and Schlumprecht extended these ideas to infinite , showing that separable reflexive s avoid certain infinite weakly null trees with uniform branching control, providing a tree-metric analogue for reflexivity beyond finite approximations.

Locally Convex Space Reflexivity

Extension to locally convex spaces

A locally convex topological vector space X is reflexive if the canonical evaluation map J: X \to X'', where X'' is the strong dual of the strong dual X' of X, is a surjective topological with X''. This generalizes the Banach space case by replacing the norm topology on the bidual with the strong topology \beta(X', X), defined by on bounded subsets of X. The strong dual X' consists of all continuous linear functionals on X, equipped with the topology of on the bounded sets of X. In reflexive locally convex spaces, the original topology on X coincides with the Mackey topology \tau(X, X'), generated by the seminorms p_{\phi}(x) = |\phi(x)| for \phi \in X'. This equivalence holds because reflexivity implies that X is barrelled, meaning every closed convex balanced absorbing set (barrel) is a neighborhood of the origin, and all barrelled locally convex spaces are Mackey spaces. The concept of reflexivity was extended beyond normed spaces to general locally convex topological vector spaces by in his 1953 thesis "Sur les espaces (F) et (DF)", building on duality theory to handle non-normable settings. Unlike Banach spaces, where the dual and bidual are equipped with norm topologies, reflexivity in locally convex spaces involves continuous whose topologies may arise as inductive limits of finite-dimensional or simpler spaces, accommodating structures like Fréchet or LF-spaces without a global norm.

Characterizations of reflexivity

In locally convex spaces, reflexivity admits several equivalent characterizations in terms of topologies and geometric properties. A fundamental one involves the Mackey topology, defined as the finest locally convex topology compatible with the duality between a space X and its dual X'. Specifically, a locally convex space X is reflexive if and only if the Mackey topology \tau(X, X') on X coincides with the Mackey topology \tau(X'', X') on the bidual X'', making the canonical embedding a topological isomorphism. Another key characterization draws from compactness in dual topologies. By the Alaoglu-Bourbaki , which generalizes the Banach-Alaoglu to locally spaces, the polar of any neighborhood of the origin in X is weakly* compact in X'. Extending this, X is reflexive every closed balanced bounded subset of X is weakly compact in the \sigma(X, X'). This condition highlights the interplay between boundedness and in reflexive settings. Reflexivity also manifests in barrelled space properties. A barrel in a locally space is a closed, , balanced, and absorptive set. The space X is barrelled if every barrel is a neighborhood of the origin. Reflexivity implies that X is barrelled. For complete locally spaces, reflexivity equates to the alignment of certain topologies. In particular, a complete locally space X is reflexive the strong topology \beta(X', X) on the X'—generated by seminorms \sup_{x \in B} |f(x)| for bounded B \subset X—coincides with the Mackey topology \tau(X', X) on X'. This equivalence underscores the completeness assumption in bridging strong and uniform convergence properties. In the context of bornological spaces, where the topology is generated by the uniform structure from bounded sets, reflexivity relates to the extension of linear forms. A bornological space X is reflexive if the strong X' is barrelled. Bornologicality ensures that the on X is the finest locally topology compatible with the given dual pair, and the barrelledness of X' implies that the strong and Mackey topologies on X' coincide, leading to reflexivity. In sequentially complete locally convex spaces, reflexivity can be established using the applied to bounded families of continuous linear functionals. Specifically, if every bounded subset of the is equicontinuous, then the space satisfies the conditions for the Mackey to equal the strong , implying reflexivity. Recent results extend these conditions to LF-spaces, which are countable strict inductive limits of Fréchet spaces. For instance, a strict (LF)-space that is complete and Montel is reflexive.

Semireflexive spaces

A locally convex space X is semireflexive if the canonical mapping J: X \to X^{**} is surjective, establishing a as sets between X and its bidual X^{**}. This condition represents a weakening of full reflexivity, where the mapping is a topological , by requiring surjectivity without the continuous inverse. The concept was introduced by Grothendieck in his work on topological vector spaces. A characterization of semireflexive spaces is that X is semireflexive the \sigma(X, X') on X has the Heine-Borel property, meaning every closed and bounded subset of X is weakly compact. This property ensures that the weak compactness in the space corresponds to the surjectivity of the embedding. Semireflexive spaces relate to reflexivity in that if X is semireflexive and its X^* is also semireflexive, then X is reflexive. This follows from the fact that the surjectivity for both X and X^* implies the bijectivity and topological required for reflexivity in the locally convex setting. Examples of semireflexive spaces include Montel spaces, where bounded sets are relatively compact, implying the necessary weak compactness conditions. On the other hand, non-semireflexive examples include certain inductive limits of non-barrelled spaces, where the canonical mapping fails to be surjective due to the lack of appropriate separation properties.

Sufficient conditions for reflexivity

A locally convex is reflexive if it is both barrelled and semireflexive. This condition arises from the fact that barrelledness ensures the coincidence of the Mackey topology with the strong topology on the dual under semireflexivity, thereby making the space the strong dual of its bidual. Fréchet spaces, being complete metrizable locally spaces, are always barrelled. Thus, a Fréchet space is reflexive it is semireflexive. Montel spaces provide another class of reflexive spaces. A Montel space is a complete locally space in which every equicontinuous subset is relatively compact. Such spaces are automatically barrelled and semireflexive, hence reflexive. A locally space X is reflexive if it is bornological and its strong X^* is barrelled. Bornologicality ensures that the topology on X is the finest locally topology compatible with the given pair, and the barrelledness of X^* implies that the strong and Mackey topologies on X^* coincide, leading to reflexivity. In sequentially complete locally spaces, reflexivity can be established using the applied to bounded families of continuous linear functionals. Specifically, if every bounded subset of the dual is equicontinuous, then the space satisfies the conditions for the Mackey topology to equal the strong topology, implying reflexivity. Recent results extend these conditions to LF-spaces, which are countable strict inductive limits of Fréchet spaces. For instance, a strict (LF)-space that is complete and Montel is reflexive.

Properties of reflexive locally convex spaces

In reflexive locally convex spaces, the strong topology and the Mackey topology coincide both on the space itself and on its . This equivalence extends the corresponding property from Banach spaces to the broader setting of locally convex topologies, ensuring that the finest locally convex topology compatible with the duality pair aligns with the topology of on convex, balanced, and absorbing sets. Reflexive locally convex spaces are semi-Montel, meaning that every bounded subset is relatively compact in the . Equivalently, in the , every equicontinuous subset is relatively weakly compact, which follows from the surjectivity of the canonical embedding into the bidual and the Mackey-Arens theorem. This property distinguishes reflexive spaces by guaranteeing weak compactness for bounded sets without requiring the stronger sequential compactness of Montel spaces. Reflexivity is preserved under the formation of products, direct sums, and quotients, provided the original space is complete. For instance, the product of reflexive spaces inherits reflexivity through the identification of the bidual with the product of the biduals, while quotients by closed hyperplanes maintain the isomorphism with the bidual under completeness assumptions. A key theorem states that every reflexive locally space is fully complete, in the sense that every Cauchy net converges. This follows from the reflexivity ensuring that the space is barrelled and the on bounded sets is complete, aligning with the Banach-Dieudonné theorem generalized to locally convex settings. Unlike the Banach space case, where reflexivity of the dual follows directly from the norm structure, in general locally convex spaces, reflexivity implies that the strong dual is itself reflexive. This bidirectional reflexivity arises because the strong bidual coincides with the original space, making the strong dual's bidual isomorphic to the original space under the strong topology.

Examples and counterexamples

The \mathcal{S}(\mathbb{R}^n) of rapidly decreasing smooth functions on \mathbb{R}^n is a prototypical example of a reflexive Fréchet space. Its reflexivity follows from the fact that it is a Fréchet-Schwartz space, where bounded sets are relatively compact, implying relative weak compactness and thus reflexivity via standard criteria for locally convex spaces. Another example is the space \mathcal{D}(\Omega) of smooth test functions with compact support on an open set \Omega \subset \mathbb{R}^n, equipped with its standard inductive limit topology, forming a complete strict (LF)-space that is reflexive. The reflexivity of \mathcal{D}(\Omega) is established by showing that the canonical embedding into its bidual is an , with the on the bidual restricting to the on the image. Counterexamples to reflexivity in the category of locally convex spaces include inductive limits of non-reflexive spaces. For instance, if each space in the directed system is non-reflexive, the resulting inductive limit inherits this property and remains non-reflexive. Non-barrelled inductive limits provide further illustrations, as the failure of the barrelled property can prevent the canonical evaluation map from being surjective onto the bidual. Strict inductive limits of sequences of non-Montel Fréchet spaces often fail to be reflexive, particularly when the component spaces are themselves non-reflexive. Such constructions highlight how the inductive topology may not preserve reflexivity without additional assumptions like Montelness in the components. A non-semireflexive example arises in incomplete variants of distribution spaces, such as the space of distributions \mathcal{D}'(\Omega) endowed with a topology weaker than the standard strong dual topology, where the to the bidual is neither surjective nor a topological due to incompleteness. Modern counterexamples from the 2000s demonstrate non-reflexive bornological spaces, such as certain products of bornological spaces that fail to be bornological under set-theoretic assumptions, leading to structures where bounded sets do not align with reflexive properties. These examples underscore the role of incompleteness in bornological contexts, revealing pathologies absent in complete cases.

Broader Notions of Reflexivity

Reflexivity in Fréchet spaces

In Fréchet spaces, which are complete metrizable locally convex topological vector spaces, reflexivity admits a precise characterization: a Fréchet space is reflexive if and only if it is barrelled and semi-Montel. All Fréchet spaces are barrelled, where a barrelled space is one in which every barrel—an absorbing, balanced, convex, and closed set—is a neighborhood of the origin, ensuring the validity of the uniform boundedness principle in this context. Semi-Montel means that every bounded subset is relatively compact in the weak topology, a condition weaker than the full Montel property where bounded sets are relatively compact in the original topology but sufficient for reflexivity when combined with barrelledness in the metrizable complete setting. This equivalence highlights the interplay between topological completeness, metrizability, and bounded set behavior unique to Fréchet spaces. A notable property in this framework concerns nuclear Fréchet spaces, which admit an approximation by Hilbert spaces via tensor products. Among these, Schwartz spaces—nuclear Fréchet spaces defined by seminorms controlling derivatives and decay at , such as the classical space of test functions on \mathbb{R}^n—are reflexive. This reflexivity follows from the relative compactness of bounded sets in the , aligning with the semi-Montel condition, and underscores how decay properties enhance duality in nuclear settings. An illustrative example is the space of entire functions on \mathbb{C}, denoted H(\mathbb{C}), endowed with the Fréchet topology of on compact subsets. This space is Montel—hence barrelled and semi-Montel—rendering it reflexive, as bounded sets (families of entire functions bounded on compacts) are equicontinuous and thus relatively compact by standard estimates from .

Reflexivity in other topological vector spaces

Reflexivity in the context of non-locally convex topological vector spaces (TVS) is inherently limited because such spaces lack a basis of convex neighborhoods of the . A fundamental result states that a TVS admits a non-trivial continuous dual if and only if it possesses a proper convex neighborhood of the . Consequently, in non-locally convex TVS, the continuous dual is trivial, rendering the canonical evaluation map into the bidual degenerate and preventing reflexivity. Moreover, even if one considers algebraic duals, the strong bidual topology on any TVS is always locally convex by construction, as it is generated by seminorms of on bounded sets. Thus, reflexivity—requiring a topological with the bidual—forces the original space to be locally convex, implying that no non-trivial non-locally convex TVS can be reflexive. In bornological spaces, reflexivity is often approached through variants like B-reflexivity, where the canonical map to the bidual equipped with the bornological topology (generated by absorbing balanced sets) is considered. A bornological TVS is B-reflexive if this map is a topological . For locally convex bornological spaces, reflexivity in the standard sense coincides with B-reflexivity. More generally, the reflexivity of a bornological space is equivalent to the reflexivity of its under the associated fine locally convex , as the bornology determines bounded sets, and preserves the duality in this setting. Other notions of reflexivity arise in specialized TVS frameworks. In ordered topological vector spaces, reflexivity can be defined via duality, where the space is isomorphic to its order —the set of order-preserving linear functionals—equipped with an appropriate . For instance, an ordered linear space is reflexive if it coincides with its order bidual, ensuring the order structure is preserved under duality. Similarly, in inductive limits of TVS without assuming , reflexivity holds if the limit is almost regular, meaning every bounded closed is contained in a of a compact set from one of the steps; this condition ensures the dual behaves well despite potential incompleteness. Recent extensions in the have explored reflexivity in quasi-complete TVS, leveraging automatic theorems, which assert that bounded linear operators between certain quasi-complete spaces are continuous. In quasi-complete TVS—where every closed bounded is complete—reflexivity implies enhanced duality properties, such as the bidual map being open, facilitated by automatic continuity in non-metrizable settings. Every reflexive TVS is quasi-complete, providing a bridge to broader duality results without full completeness. A notable counterexample illustrating non-reflexivity occurs in non-Hausdorff TVS, such as the indiscrete on an infinite-dimensional , where the only continuous linear functionals are zero, yielding a trivial continuous and thus a degenerate bidual . In such cases, the space fails to separate points topologically, precluding any with its bidual.