In the area of mathematics known as functional analysis, a **reflexive space** is a locally convex topological vector space (TVS) such that the canonical evaluation map from into its bidual (which is the strong dual of the strong dual of ) is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space (and so in particular, every Banach space) is reflexive if and only if the canonical evaluation map from into its bidual is surjective; in this case the normed space is necessarily also a Banach space. In 1951, R. C. James discovered a Banach space, now known as James' space, that is *not* reflexive but is nevertheless isometrically isomorphic to its bidual (any such isomorphism is thus necessarily *not* the canonical evaluation map).

- Definition
- Reflexive Banach spaces
- Remark
- Examples
- Properties
- Super-reflexive space
- Finite trees in Banach spaces
- Reflexive locally convex spaces
- Semireflexive spaces
- Characterizations of reflexive spaces
- Sufficient conditions
- Properties 2
- Examples 2
- Other types of reflexivity
- See also
- Notes
- Citations
- References

Reflexive spaces play an important role in the general theory of locally convex TVSs and in the theory of Banach spaces in particular. Hilbert spaces are prominent examples of reflexive Banach spaces. Reflexive Banach spaces are often characterized by their geometric properties.

- Definition of the bidual

Suppose that is a topological vector space (TVS) over the field (which is either the real or complex numbers) whose continuous dual space, **separates points** on (that is, for any there exists some such that ). Let and both denote the strong dual of which is the vector space of continuous linear functionals on endowed with the topology of uniform convergence on bounded subsets of ; this topology is also called the **strong dual topology** and it is the "default" topology placed on a continuous dual space (unless another topology is specified). If is a normed space, then the strong dual of is the continuous dual space with its usual norm topology. The **bidual** of denoted by is the strong dual of ; that is, it is the space ^{ [1] } If is a normed space, then is the continuous dual space of the Banach space with its usual norm topology.

- Definitions of the evaluation map and reflexive spaces

For any let be defined by where is a linear map called the **evaluation map at **; since is necessarily continuous, it follows that Since separates points on the linear map defined by is injective where this map is called the **evaluation map** or the **canonical map**. Call ** semi-reflexive ** if is bijective (or equivalently, surjective) and we call **reflexive** if in addition is an isomorphism of TVSs.^{ [1] } A normable space is reflexive if and only if it is semi-reflexive or equivalently, if and only if the evaluation map is surjective.

Suppose is a normed vector space over the number field or (the real numbers or the complex numbers), with a norm Consider its dual normed space that consists of all continuous linear functionals and is equipped with the dual norm defined by

The dual is a normed space (a Banach space to be precise), and its dual normed space is called **bidual space** for The bidual consists of all continuous linear functionals and is equipped with the norm dual to Each vector generates a scalar function by the formula:

and is a continuous linear functional on that is, One obtains in this way a map

called **evaluation map**, that is linear. It follows from the Hahn–Banach theorem that is injective and preserves norms:

that is, maps isometrically onto its image in Furthermore, the image is closed in but it need not be equal to

A normed space is called **reflexive** if it satisfies the following equivalent conditions:

- the evaluation map is surjective,
- the evaluation map is an isometric isomorphism of normed spaces,
- the evaluation map is an isomorphism of normed spaces.

A reflexive space is a Banach space, since is then isometric to the Banach space

A Banach space is reflexive if it is linearly isometric to its bidual under this canonical embedding James' space is an example of a non-reflexive space which is linearly isometric to its bidual. Furthermore, the image of James' space under the canonical embedding has codimension one in its bidual. ^{ [2] } A Banach space is called **quasi-reflexive** (of order ) if the quotient has finite dimension

- Every finite-dimensional normed space is reflexive, simply because in this case, the space, its dual and bidual all have the same linear dimension, hence the linear injection from the definition is bijective, by the rank–nullity theorem.
- The Banach space of scalar sequences tending to 0 at infinity, equipped with the supremum norm, is not reflexive. It follows from the general properties below that and are not reflexive, because is isomorphic to the dual of and is isomorphic to the dual of
- All Hilbert spaces are reflexive, as are the Lp spaces for More generally: all uniformly convex Banach spaces are reflexive according to the Milman–Pettis theorem. The and spaces are not reflexive (unless they are finite dimensional, which happens for example when is a measure on a finite set). Likewise, the Banach space of continuous functions on is not reflexive.
- The spaces of operators in the Schatten class on a Hilbert space are uniformly convex, hence reflexive, when When the dimension of is infinite, then (the trace class) is not reflexive, because it contains a subspace isomorphic to and (the bounded linear operators on ) is not reflexive, because it contains a subspace isomorphic to In both cases, the subspace can be chosen to be the operators diagonal with respect to a given orthonormal basis of

If a Banach space is isomorphic to a reflexive Banach space then is reflexive.^{ [3] }

Every closed linear subspace of a reflexive space is reflexive. The continuous dual of a reflexive space is reflexive. Every quotient of a reflexive space by a closed subspace is reflexive.^{ [4] }

Let be a Banach space. The following are equivalent.

- The space is reflexive.
- The continuous dual of is reflexive.
^{ [5] } - The closed unit ball of is compact in the weak topology. (This is known as Kakutani's Theorem.)
^{ [6] } - Every bounded sequence in has a weakly convergent subsequence.
^{ [7] } - Every continuous linear functional on attains its supremum on the closed unit ball in
^{ [8] }(James' theorem)

Since norm-closed convex subsets in a Banach space are weakly closed,^{ [9] } it follows from the third property that closed bounded convex subsets of a reflexive space are weakly compact. Thus, for every decreasing sequence of non-empty closed bounded convex subsets of the intersection is non-empty. As a consequence, every continuous convex function on a closed convex subset of such that the set

is non-empty and bounded for some real number attains its minimum value on

The promised geometric property of reflexive Banach spaces is the following: if is a closed non-empty convex subset of the reflexive space then for every there exists a such that minimizes the distance between and points of This follows from the preceding result for convex functions, applied to Note that while the minimal distance between and is uniquely defined by the point is not. The closest point is unique when is uniformly convex.

A reflexive Banach space is separable if and only if its continuous dual is separable. This follows from the fact that for every normed space separability of the continuous dual implies separability of ^{ [10] }

Informally, a super-reflexive Banach space has the following property: given an arbitrary Banach space if all finite-dimensional subspaces of have a very similar copy sitting somewhere in then must be reflexive. By this definition, the space itself must be reflexive. As an elementary example, every Banach space whose two dimensional subspaces are isometric to subspaces of satisfies the parallelogram law, hence^{ [11] } is a Hilbert space, therefore is reflexive. So is super-reflexive.

The formal definition does not use isometries, but almost isometries. A Banach space is **finitely representable**^{ [12] } in a Banach space if for every finite-dimensional subspace of and every there is a subspace of such that the multiplicative Banach–Mazur distance between and satisfies

A Banach space finitely representable in is a Hilbert space. Every Banach space is finitely representable in The Lp space is finitely representable in

A Banach space is **super-reflexive** if all Banach spaces finitely representable in are reflexive, or, in other words, if no non-reflexive space is finitely representable in The notion of ultraproduct of a family of Banach spaces^{ [13] } allows for a concise definition: the Banach space is super-reflexive when its ultrapowers are reflexive.

James proved that a space is super-reflexive if and only if its dual is super-reflexive.^{ [12] }

One of James' characterizations of super-reflexivity uses the growth of separated trees.^{ [14] } The description of a vectorial binary tree begins with a rooted binary tree labeled by vectors: a tree of height in a Banach space is a family of vectors of that can be organized in successive levels, starting with level 0 that consists of a single vector the root of the tree, followed, for by a family of 2 vectors forming level

that are the children of vertices of level In addition to the tree structure, it is required here that each vector that is an internal vertex of the tree be the midpoint between its two children:

Given a positive real number the tree is said to be **-separated** if for every internal vertex, the two children are -separated in the given space norm:

Theorem.^{ [14] }The Banach space is super-reflexive if and only if for every there is a number such that every -separated tree contained in the unit ball of has height less than

Uniformly convex spaces are super-reflexive.^{ [14] } Let be uniformly convex, with modulus of convexity and let be a real number in By the properties of the modulus of convexity, a -separated tree of height contained in the unit ball, must have all points of level contained in the ball of radius By induction, it follows that all points of level are contained in the ball of radius

If the height was so large that

then the two points of the first level could not be -separated, contrary to the assumption. This gives the required bound function of only.

Using the tree-characterization, Enflo proved^{ [15] } that super-reflexive Banach spaces admit an equivalent uniformly convex norm. Trees in a Banach space are a special instance of vector-valued martingales. Adding techniques from scalar martingale theory, Pisier improved Enflo's result by showing^{ [16] } that a super-reflexive space admits an equivalent uniformly convex norm for which the modulus of convexity satisfies, for some constant and some real number

The notion of reflexive Banach space can be generalized to topological vector spaces in the following way.

Let be a topological vector space over a number field (of real numbers or complex numbers ). Consider its strong dual space which consists of all continuous linear functionals and is equipped with the strong topology that is,, the topology of uniform convergence on bounded subsets in The space is a topological vector space (to be more precise, a locally convex space), so one can consider its strong dual space which is called the **strong bidual space** for It consists of all continuous linear functionals and is equipped with the strong topology Each vector generates a map by the following formula:

This is a continuous linear functional on that is,, This induces a map called the **evaluation map**:

This map is linear. If is locally convex, from the Hahn–Banach theorem it follows that is injective and open (that is, for each neighbourhood of zero in there is a neighbourhood of zero in such that ). But it can be non-surjective and/or discontinuous.

A locally convex space is called

**semi-reflexive**if the evaluation map is surjective (hence bijective),**reflexive**if the evaluation map is surjective and continuous (in this case is an isomorphism of topological vector spaces^{ [17] }).

**Theorem ^{ [18] }** — A locally convex Hausdorff space is semi-reflexive if and only if with the -topology has the Heine–Borel property (i.e. weakly closed and bounded subsets of are weakly compact).

**Theorem ^{ [19] }^{ [20] }** — A locally convex space is reflexive if and only if it is semi-reflexive and barreled.

**Theorem ^{ [21] }** — The strong dual of a semireflexive space is barrelled.

**Theorem ^{ [22] }** — If is a Hausdorff locally convex space then the canonical injection from into its bidual is a topological embedding if and only if is infrabarreled.

If is a Hausdorff locally convex space then the following are equivalent:

- is semireflexive;
- The weak topology on had the Heine-Borel property (that is, for the weak topology every closed and bounded subset of is weakly compact).
^{ [1] } - If linear form on that continuous when has the strong dual topology, then it is continuous when has the weak topology;
^{ [23] } - is barreled;
^{ [23] } - weak the weak topology is quasi-complete.
^{ [23] }

If is a Hausdorff locally convex space then the following are equivalent:

- is reflexive;
- is semireflexive and infrabarreled;
^{ [22] } - is semireflexive and barreled;
- is barreled and the weak topology on had the Heine-Borel property (that is, for the weak topology every closed and bounded subset of is weakly compact).
^{ [1] } - is semireflexive and quasibarrelled.
^{ [24] }

If is a normed space then the following are equivalent:

- is reflexive;
- The closed unit ball is compact when has the weak topology
^{ [25] } - is a Banach space and is reflexive.
^{ [26] } - Every sequence with for all of nonempty closed bounded convex subsets of has nonempty intersection.
^{ [27] }

**Theorem ^{ [28] }** — A real Banach space is reflexive if and only if every pair of non-empty disjoint closed convex subsets, one of which is bounded, can be strictly separated by a hyperplane.

** James' theorem ** — A Banach space is reflexive if and only if every continuous linear functional on attains its supremum on the closed unit ball in

- Normed spaces

A normed space that is semireflexive is a reflexive Banach space.^{ [29] } A closed vector subspace of a reflexive Banach space is reflexive.^{ [22] }

Let be a Banach space and a closed vector subspace of If two of and are reflexive then they all are.^{ [22] } This is why reflexivity is referred to as a *three-space property*.^{ [22] }

- Topological vector spaces

If a barreled locally convex Hausdorff space is semireflexive then it is reflexive.^{ [1] }

The strong dual of a reflexive space is reflexive.^{ [30] } The strong dual of a reflexive space is reflexive.^{ [30] } Every Montel space is reflexive.^{ [25] } And the strong dual of a Montel space is a Montel space (and thus is reflexive).^{ [25] }

A locally convex Hausdorff reflexive space is barrelled. If is a normed space then is an isometry onto a closed subspace of ^{ [29] } This isometry can be expressed by:

Suppose that is a normed space and is its bidual equipped with the bidual norm. Then the unit ball of is dense in the unit ball of for the weak topology ^{ [29] }

- Every finite-dimensional Hausdorff topological vector space is reflexive, because is bijective by linear algebra, and because there is a unique Hausdorff vector space topology on a finite dimensional vector space.
- A normed space is reflexive as a normed space if and only if it is reflexive as a locally convex space. This follows from the fact that for a normed space its dual normed space coincides as a topological vector space with the strong dual space As a corollary, the evaluation map coincides with the evaluation map and the following conditions become equivalent:
- is a reflexive normed space (that is, is an isomorphism of normed spaces),
- is a reflexive locally convex space (that is, is an isomorphism of topological vector spaces
^{ [17] }), - is a semi-reflexive locally convex space (that is, is surjective).

- A (somewhat artificial) example of a semi-reflexive space that is not reflexive is obtained as follows: let be an infinite dimensional reflexive Banach space, and let be the topological vector space that is, the vector space equipped with the weak topology. Then the continuous dual of and are the same set of functionals, and bounded subsets of (that is, weakly bounded subsets of ) are norm-bounded, hence the Banach space is the strong dual of Since is reflexive, the continuous dual of is equal to the image of under the canonical embedding but the topology on (the weak topology of ) is not the strong topology that is equal to the norm topology of
- Montel spaces are reflexive locally convex topological vector spaces. In particular, the following functional spaces frequently used in functional analysis are reflexive locally convex spaces:
^{ [31] }- the space of smooth functions on arbitrary (real) smooth manifold and its strong dual space of distributions with compact support on
- the space of smooth functions with compact support on arbitrary (real) smooth manifold and its strong dual space of distributions on
- the space of holomorphic functions on arbitrary complex manifold and its strong dual space of analytic functionals on
- the Schwartz space on and its strong dual space of tempered distributions on

- Counter-examples

- There exists a non-reflexive locally convex TVS whose strong dual is reflexive.
^{ [32] }

A stereotype space, or polar reflexive space, is defined as a topological vector space (TVS) satisfying a similar condition of reflexivity, but with the topology of uniform convergence on totally bounded subsets (instead of bounded subsets) in the definition of dual space More precisely, a TVS is called polar reflexive^{ [33] } or stereotype if the evaluation map into the second dual space

is an isomorphism of topological vector spaces.^{ [17] } Here the stereotype dual space is defined as the space of continuous linear functionals endowed with the topology of uniform convergence on totally bounded sets in (and the *stereotype second dual space* is the space dual to in the same sense).

In contrast to the classical reflexive spaces the class **Ste** of stereotype spaces is very wide (it contains, in particular, all Fréchet spaces and thus, all Banach spaces), it forms a closed monoidal category, and it admits standard operations (defined inside of **Ste**) of constructing new spaces, like taking closed subspaces, quotient spaces, projective and injective limits, the space of operators, tensor products, etc. The category **Ste** have applications in duality theory for non-commutative groups.

Similarly, one can replace the class of bounded (and totally bounded) subsets in in the definition of dual space by other classes of subsets, for example, by the class of compact subsets in – the spaces defined by the corresponding reflexivity condition are called *reflective*,^{ [34] }^{ [35] } and they form an even wider class than **Ste**, but it is not clear (2012), whether this class forms a category with properties similar to those of **Ste**.

- Grothendieck space
- A generalization which has some of the properties of reflexive spaces and includes many spaces of practical importance is the concept of Grothendieck space.

- Reflexive operator algebra

- 1 2 3 4 5 Trèves 2006, pp. 372–374.
- ↑ R. C. James (1951). "A non-reflexive Banach space isometric with its second conjugate space".
*Proc. Natl. Acad. Sci. U.S.A*.**37**(3): 174–177. Bibcode:1951PNAS...37..174J. doi: 10.1073/pnas.37.3.174 . PMC 1063327 . PMID 16588998. - ↑ Proposition 1.11.8 in Megginson (1998 , p. 99).
- ↑ Megginson (1998 , pp. 104–105).
- ↑ Corollary 1.11.17, p. 104 in Megginson (1998).
- ↑ Conway 1985, Theorem V.4.2, p. 135.
- ↑ Since weak compactness and weak sequential compactness coincide by the Eberlein–Šmulian theorem.
- ↑ Theorem 1.13.11 in Megginson (1998 , p. 125).
- ↑ Theorem 2.5.16 in Megginson (1998 , p. 216).
- ↑ Theorem 1.12.11 and Corollary 1.12.12 in Megginson (1998 , pp. 112–113).
- ↑ see this characterization of Hilbert space among Banach spaces
- 1 2 James, Robert C. (1972), "Super-reflexive Banach spaces", Can. J. Math.
**24**:896–904. - ↑ Dacunha-Castelle, Didier; Krivine, Jean-Louis (1972), "Applications des ultraproduits à l'étude des espaces et des algèbres de Banach" (in French), Studia Math.
**41**:315–334. - 1 2 3 see James (1972).
- ↑ Enflo, Per (1973), "Banach spaces which can be given an equivalent uniformly convex norm", Israel Journal of Mathematics
**13**:281–288. - ↑ Pisier, Gilles (1975), "Martingales with values in uniformly convex spaces", Israel Journal of Mathematics
**20**:326–350. - 1 2 3 An
*isomorphism of topological vector spaces*is a linear and a homeomorphic map - ↑ Edwards 1965, 8.4.2.
- ↑ Schaefer 1966, 5.6, 5.5.
- ↑ Edwards 1965, 8.4.5.
- ↑ Edwards 1965, 8.4.3.
- 1 2 3 4 5 Narici & Beckenstein 2011, pp. 488–491.
- 1 2 3 Schaefer & Wolff 1999, p. 144.
- ↑ Khaleelulla 1982, pp. 32–63.
- 1 2 3 Trèves 2006, p. 376.
- ↑ Trèves 2006, p. 377.
- ↑ Bernardes Jr. 2012.
- ↑ Narici & Beckenstein 2011, pp. 212.
- 1 2 3 Trèves 2006, p. 375.
- 1 2 Schaefer & Wolff 1999, p. 145.
- ↑ Edwards 1965 , 8.4.7.
- ↑ Schaefer & Wolff 1999, pp. 190–202.
- ↑ Köthe, Gottfried (1983).
*Topological Vector Spaces I*. Springer Grundlehren der mathematischen Wissenschaften. Springer. ISBN 978-3-642-64988-2. - ↑ Garibay Bonales, F.; Trigos-Arrieta, F. J.; Vera Mendoza, R. (2002). "A characterization of Pontryagin-van Kampen duality for locally convex spaces".
*Topology and Its Applications*.**121**(1–2): 75–89. doi: 10.1016/s0166-8641(01)00111-0 . - ↑ Akbarov, S. S.; Shavgulidze, E. T. (2003). "On two classes of spaces reflexive in the sense of Pontryagin".
*Mat. Sbornik*.**194**(10): 3–26.

In mathematics, more specifically in functional analysis, a **Banach space** is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

In mathematics, any vector space * has a corresponding ***dual vector space** consisting of all linear forms on *, together with the vector space structure of pointwise addition and scalar multiplication by constants.*

The **Hahn–Banach theorem** is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting". Another version of the Hahn–Banach theorem is known as the **Hahn–Banach separation theorem** or the hyperplane separation theorem, and has numerous uses in convex geometry.

In mathematics, a **normed vector space** or **normed space** is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" in the real world. A norm is a real-valued function defined on the vector space that is commonly denoted and has the following properties:

- It is nonnegative, that is for every vector x, one has
- It is positive on nonzero vectors, that is,
- For every vector x, and every scalar one has
- The triangle inequality holds; that is, for every vectors x and y, one has

In mathematics, **weak topology** is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis.

In mathematics, a **topological vector space** is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

In functional analysis and related areas of mathematics, **Fréchet spaces**, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically *not* Banach spaces.

In functional analysis and related areas of mathematics, **locally convex topological vector spaces** (**LCTVS**) or **locally convex spaces** are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

In functional analysis and related branches of mathematics, the **Banach–Alaoglu theorem** states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak-* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.

In functional analysis and related areas of mathematics, a **sequence space** is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field *K* of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in *K*, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.

In functional analysis and related areas of mathematics, a **barrelled space** is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A **barrelled set** or a **barrel** in a topological vector space is a set that is convex, balanced, absorbing, and closed. Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them.

In mathematics, particularly in functional analysis, a **bornological space** is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces are distinguished by that property that a linear map from a bornological space into any locally convex spaces is continuous if and only if it is a bounded linear operator.

In mathematics, **nuclear spaces** are topological vector spaces that can be viewed as a generalization of finite dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces are however quite different from Hilbert spaces, another generalization of finite dimensional Euclidean spaces. They were introduced by Alexander Grothendieck.

In the area of mathematics known as functional analysis, a **semi-reflexive space** is a locally convex topological vector space (TVS) *X* such that the canonical evaluation map from *X* into its bidual is bijective. If this map is also an isomorphism of TVSs then it is called **reflexive**.

The strongest locally convex topological vector space (TVS) topology on the tensor product of two locally convex TVSs, making the canonical map continuous is called the **projective topology** or the **π-topology**. When is endowed with this topology then it is denoted by and called the **projective tensor product** of and

In mathematics, the **injective tensor product** of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so its completion is called the *completed injective tensor products*. Injective tensor products have applications outside of nuclear spaces. In particular, as described below, up to TVS-isomorphism, many TVSs that are defined for real or complex valued functions, for instance, the Schwartz space or the space of continuously differentiable functions, can be immediately extended to functions valued in a Hausdorff locally convex TVS Y with*out* any need to extend definitions from real/complex-valued functions to Y-valued functions.

In functional analysis and related areas of mathematics, the **strong dual space** of a topological vector space (TVS) is the continuous dual space of equipped with the **strong** (**dual**) **topology** or the **topology of uniform convergence on bounded subsets of ** where this topology is denoted by or The coarsest polar topology is called weak topology. The strong dual space plays such an important role in modern functional analysis, that the continuous dual space is usually assumed to have the strong dual topology unless indicated otherwise. To emphasize that the continuous dual space, has the strong dual topology, or may be written.

In mathematics, specifically in order theory and functional analysis, a **locally convex vector lattice (LCVL)** is a topological vector lattice that is also a locally convex space. LCVLs are important in the theory of topological vector lattices.

In functional analysis and related areas of mathematics, **distinguished spaces** are topological vector spaces (TVSs) having the property that weak-* bounded subsets of their biduals are contained in the weak-* closure of some bounded subset of the bidual.

In functional analysis and related areas of mathematics, a **metrizable** topological vector space (TVS) is a TVS whose topology is induced by a metric. An **LM-space** is an inductive limit of a sequence of locally convex metrizable TVS.

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