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| In [[functional analysis]], a [[Banach space]] (or more generally a [[locally convex topological vector space]]) is called '''reflexive''' if it coincides with the [[Dual space#Continuous dual space|continuous dual]] of its continuous dual space, both as linear space and as topological space. Reflexive Banach spaces are often characterized by their geometric properties.
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| == Reflexive Banach spaces ==
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| Suppose <math>X</math> is a [[normed vector space]] over the number field <math>\mathbb{F} = \mathbb{R}</math> or <math>\mathbb{F} = \mathbb {C}</math> (the [[real number|real]] or [[complex number]]s), with a norm <math>\|\cdot\|</math>. Consider its [[dual norm|dual normed space]] <math>X'</math>, that consists of all [[continuous function|continuous]] [[linear functional]]s <math>f:X\to {\mathbb F}</math> and is equipped with the [[dual norm]] <math>\|\cdot\|'</math> defined by
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| :<math>\|f\|' = \sup \{ |f(x)| \,:\, x \in X, \ \|x\| \le 1 \}.</math>
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| The dual <math>X'</math> is a normed space (a [[Banach space]] to be precise), and its dual normed space <math>X''=(X')'</math> is called '''bidual space''' for <math>X</math>. The bidual consists of all continuous linear functionals <math>h:X'\to {\mathbb F}</math> and is equipped with the norm <math>\|\cdot\|''</math> dual to <math>\|\cdot\|'</math>. Each vector <math>x\in X</math> generates a scalar function <math>J(x):X'\to{\mathbb F}</math> by the formula:
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| :<math>
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| J(x)(f)=f(x),\qquad f\in X',
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| </math>
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| and <math>J(x)</math> is a continuous linear functional on <math>X'</math>, ''i.e.'', <math>J(x)\in X''</math>. One obtains in this way a map
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| :<math> J: X \to X'' </math>
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| called '''evaluation map''', that is linear. It follows from the [[Hahn–Banach theorem]] that <math>J</math> is injective and preserves norms:
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| :<math>
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| \forall x\in X\qquad \|J(x)\|''=\|x\|,
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| </math>
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| ''i.e.'', <math>J</math> maps <math>X</math> isometrically onto its image <math>J(X)</math> in <math>X''</math>. The image <math>J(X)</math> need not be equal to <math>X''</math>.
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| A normed space <math>X</math> is called '''reflexive''' if it satisfies the following equivalent conditions:
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| :(i) the evaluation map <math>J:X\to X''</math> is surjective,
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| :(ii) the evaluation map <math>J:X\to X''</math> is an [[Banach space|isometric isomorphism]] of normed spaces,
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| :(iii) the evaluation map <math>J:X\to X''</math> is an [[Banach space|isomorphism]] of normed spaces.
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| A reflexive space <math>X</math> is a Banach space, since <math>X</math> is then isometric to the Banach space <math>X''</math>.
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| === Remark ===
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| A Banach space ''X'' is reflexive if it is linearly isometric to its bidual under this canonical embedding ''J'', but it has been shown that there exists a non-reflexive space ''X'' which is linearly isometric to {{nowrap|''X'' ′′}}. Furthermore, the image ''J''(''X'') of this space has codimension one in its bidual.
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| <ref>{{cite journal
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| | author = R. C. James
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| | title = A non-reflexive Banach space isometric with its second conjugate space
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| | journal = Proc. Natl. Acad. Sci. U.S.A.
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| | volume = 37
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| | pages = 174–177
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| | year = 1951 }}</ref>
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| A Banach space ''X'' is called '''quasi-reflexive''' (of order ''d'') if the quotient {{nowrap|''X'' ′′ / ''J''(''X'')}} has finite dimension ''d''.
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| === Examples ===
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| 1) 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 ''J'' from the definition is bijective, by the [[rank-nullity theorem]].
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| 2) The Banach space ''c''<sub>0</sub> of scalar sequences tending to 0 at infinity, equipped with the supremum norm, is not reflexive. It follows from the general properties below that ℓ<sup>1</sup> and ℓ<sup>∞</sup> are not reflexive, because ℓ<sup>1</sup> is isomorphic to the dual of ''c''<sub>0</sub>, and ℓ<sup>∞</sup> is isomorphic to the dual of ℓ<sup>1</sup>.
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| 3) All [[Hilbert space]]s are reflexive, as are the [[Lp space|''L''<sup>''p''</sup> spaces]] for {{nowrap|1 < ''p'' < ∞}}. More generally: all [[uniformly convex space|uniformly convex]] Banach spaces are reflexive according to the [[Milman–Pettis theorem]]. The ''L''<sup>1</sup>(''μ'') and ''L''<sup>∞</sup>(''μ'') spaces are not reflexive (unless they are finite dimensional, which happens for example when ''μ'' is a measure on a finite set). Likewise, the Banach space ''C''([0, 1]) of continuous functions on [0, 1] is not reflexive.
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| 4) The spaces ''S''<sub>''p''</sub>(''H'') of operators in the [[Schatten class operator|Schatten class]] on a Hilbert space ''H'' are uniformly convex, hence reflexive, when {{nowrap|1 < ''p'' < ∞}}. When the dimension of ''H'' is infinite, then ''S''<sub>1</sub>(''H'') (the [[trace class]]) is not reflexive, because it contains a subspace isomorphic to ℓ<sup>1</sup>, and ''S''<sub>∞</sub>(''H'') = ''L''(''H'') (the bounded linear operators on ''H'') is not reflexive, because it contains a subspace isomorphic to ℓ<sup>∞</sup>. In both cases, the subspace can be chosen to be the operators diagonal with respect to a given orthonormal basis of ''H''.
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| === Properties ===
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| In this section, ''dual'' means ''continuous dual''.
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| If a Banach space ''Y'' is isomorphic to a reflexive Banach space ''X'', then ''Y'' is reflexive.<ref>Proposition 1.11.8, p. 99 in {{harvtxt|Megginson|1998}}.</ref> | |
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| Every [[Closed set|closed]] [[linear subspace]] of a reflexive space is reflexive. The dual of a reflexive space is reflexive. Every [[Quotient space (linear algebra)|quotient]] of a reflexive space by a closed subspace is reflexive.<ref>pp. 104–105 in {{harvtxt|Megginson|1998}}.</ref>
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| Let ''X'' be a Banach space. The following are equivalent.
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| # The space ''X'' is reflexive.
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| # The dual of ''X'' is reflexive.<ref>Corollary 1.11.17, p. 104 in {{harvtxt|Megginson|1998}}.</ref>
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| # The closed unit ball of ''X'' is [[compact space|compact]] in the [[weak topology]]. (This is known as Kakutani's Theorem.)<ref>Conway, Theorem V.4.2, p. 135.</ref>
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| # Every bounded sequence in ''X'' has a weakly convergent subsequence.<ref>Since weak compactness and weak sequential compactness coincide by the [[Eberlein–Šmulian theorem]].</ref>
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| # Every continuous linear functional on ''X'' attains its maximum on the closed unit ball in ''X''.<ref>Theorem 1.13.11, p. 125 in {{harvtxt|Megginson|1998}}.</ref> ([[James' theorem]])
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| Since norm-closed [[Convex set|convex subsets]] in a Banach space are weakly closed,<ref>Theorem 2.5.16, p. 216 in {{harvtxt|Megginson|1998}}.</ref>
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| it follows from the third property that closed bounded convex subsets of a reflexive space ''X'' are weakly compact. Thus, for every decreasing sequence of non-empty closed bounded convex subsets of ''X'', the intersection is non-empty. As a consequence, every continuous [[convex function]] ''f'' on a closed convex subset ''C'' of ''X'', such that the set
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| :<math> C_t = \{ x \in C \,:\, f(x) \le t\} </math>
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| is non-empty and bounded for some real number ''t'', attains its minimum value on ''C''.
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| The promised geometric property of reflexive Banach spaces is the following: if ''C'' is a closed non-empty [[Convex set|convex]] subset of the reflexive space ''X'', then for every ''x'' in ''X'' there exists a ''c'' in ''C'' such that {{nowrap|ǁ''x'' − ''c''ǁ}} minimizes the distance between ''x'' and points of ''C''. This follows from the preceding result for convex functions, applied to {{nowrap|''f''(''y'') {{=}} ǁ''y'' − ''x''ǁ}}. Note that while the minimal distance between ''x'' and ''C'' is uniquely defined by ''x'', the point ''c'' is not. The closest point ''c'' is unique when ''X'' is uniformly convex.
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| A reflexive Banach space is [[separable space|separable]] if and only if its dual is separable. This follows from the fact that for every normed space ''Y'', separability of the dual {{nowrap|''Y'' ′}} implies separability {{nowrap|of ''Y''}}.<ref>Theorem 1.12.11, p. 112 and Corollary 1.12.12, p. 113 in {{harvtxt|Megginson|1998}}.</ref>
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| === Super-reflexive space ===
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| Informally, a super-reflexive Banach space ''X'' has the following property: given an arbitrary Banach space ''Y'', if all finite-dimensional subspaces of ''Y'' have a very similar copy sitting somewhere in ''X'', then ''Y'' must be reflexive. By this definition, the space ''X'' itself must be reflexive. As an elementary example, every Banach space ''Y'' whose two dimensional subspaces are [[Isometry|isometric]] to subspaces of {{nowrap|''X'' {{=}} ℓ<sup>2</sup>}} satisfies the [[parallelogram law]], hence<ref>see this [[Banach space#Characterizations of Hilbert space among Banach spaces|characterization of Hilbert space among Banach spaces]]</ref>
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| ''Y'' is a Hilbert space, therefore ''Y'' is reflexive. So ℓ<sup>2</sup> is super-reflexive.
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| The formal definition does not use isometries, but almost isometries. A Banach space ''Y'' is '''finitely representable'''<ref name="SRBS">James, Robert C. (1972), "Super-reflexive Banach spaces", Canad. J. Math. '''24''':896–904.</ref>
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| in a Banach space ''X'' if for every finite-dimensional subspace ''Y''<sub>0</sub> of ''Y'' and every {{nowrap| ε > 0}}, there is a subspace ''X''<sub>0</sub> of ''X'' such that the multiplicative [[Banach–Mazur compactum|Banach–Mazur distance]] between ''X''<sub>0</sub> and ''Y''<sub>0</sub> satisfies
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| :<math>d(X_0, Y_0) < 1 + \varepsilon.</math> | |
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| A Banach space finitely representable in ℓ<sup>2</sup> is a Hilbert space. Every Banach space is finitely representable in ''c''<sub>0</sub>. The space [[Lp space|''L''<sup>''p''</sup>([0, 1])]] is finitely representable in ℓ<sup>''p''</sup>.
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| A Banach space ''X'' is '''super-reflexive''' if all Banach spaces ''Y'' finitely representable in ''X'' are reflexive, or, in other words, if no non-reflexive space ''Y'' is finitely representable in ''X''. The notion of [[ultraproduct]] of a family of Banach spaces<ref>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.</ref>
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| allows for a concise definition: the Banach space ''X'' is super-reflexive when its ultrapowers are reflexive.
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| James proved that a space is super-reflexive if and only if its dual is super-reflexive.<ref name="SRBS" />
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| === Finite trees in Banach spaces ===
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| One of James' characterizations of super-reflexivity uses the growth of separated trees.<ref name="Tree">see {{harvtxt|James|1972}}.</ref>
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| The description of a vectorial binary tree begins with a [[Tree (graph theory)#Definitions|rooted binary tree]] labeled by vectors: a tree of [[Binary tree#Definitions for rooted trees|height]] ''n'' in a Banach space ''X'' is a family of {{nowrap| 2<sup>''n'' + 1</sup> − 1}} vectors of ''X'', that can be organized in successive levels, starting with level 0 that consists of a single vector ''x''<sub>∅</sup>, the [[Tree (graph theory)#Definitions|root]] of the tree, followed, for {{nowrap|''k'' {{=}} 1, …, ''n''}}, by a family of 2<sup>''k''</sup> vectors forming level ''k'': | |
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| :<math>\{x_{\varepsilon_1, \ldots, \varepsilon_k}\},
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| \quad \varepsilon_j = \pm 1, \quad j = 1, \ldots, k,</math>
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| that are the [[Tree (graph theory)#Definitions|children]] of vertices of level {{nowrap|''k'' − 1}}. In addition to the [[Tree (graph theory)|tree structure]], it is required here that each vector that is an [[Tree (graph theory)#Definitions|internal vertex]] of the tree be the midpoint between its two children:
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| :<math>x_\emptyset = \frac{x_1 + x_{-1}}{2}, \quad x_{\varepsilon_1, \ldots, \varepsilon_k} = \frac{x_{\varepsilon_1, \ldots, \varepsilon_k, 1} + x_{\varepsilon_1, \ldots, \varepsilon_k, -1}} {2}, \quad 1 \le k < n.</math>
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| Given a positive real number ''t'', the tree is said to be ''' ''t''-separated''' if for every internal vertex, the two children are ''t''-separated in the given space norm:
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| :<math> \|x_1 - x_{-1}\| \ge t, \quad \|x_{\varepsilon_1, \ldots, \varepsilon_k, 1} - x_{\varepsilon_1, \ldots, \varepsilon_k, -1}\| \ge t, \quad 1 \le k < n.</math>
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| <blockquote>'''Theorem.'''<ref name="Tree" />
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| The Banach space ''X'' is super-reflexive if and only if for every {{nowrap|''t'' ∈ (0, 2]}}, there is a number ''n''(''t'') such that every ''t''-separated tree contained in the unit ball of ''X'' has height less than ''n''(''t'').</blockquote>
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| [[Uniformly convex space]]s are super-reflexive.<ref name="Tree" />
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| Let ''X'' be uniformly convex, with [[Modulus and characteristic of convexity|modulus of convexity]] δ<sub>''X''</sub> and let ''t'' be a real number in {{nowrap|(0, 2]}}. By the [[Modulus and characteristic of convexity#Definitions|properties]] of the modulus of convexity, a ''t''-separated tree of height ''n'', contained in the unit ball, must have all points of level {{nowrap|''n'' − 1}} contained in the ball of radius {{nowrap|1 − δ<sub>''X''</sub>(''t'') < 1}}. By induction, it follows that all points of level {{nowrap|''n'' − ''j''}} are contained in the ball of radius
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| :<math> (1 - \delta_X(t))^{j}, \ j = 1, \ldots, n.</math>
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| If the height ''n'' was so large that
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| :<math> (1 - \delta_X(t))^{n-1} < t / 2, </math>
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| then the two points ''x''<sub>1</sub>, ''x''<sub>−1</sub> of the first level could not be ''t''-separated, contrary to the assumption. This gives the required bound ''n''(''t''), function of δ<sub>''X''</sub>(''t'') only. | |
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| Using the tree-characterization, [[Per Enflo|Enflo]] proved<ref>Enflo, Per (1973), "Banach spaces which can be given an equivalent uniformly convex norm", Israel J. Math. '''13''':281–288.</ref>
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| that super-reflexive Banach spaces admit an equivalent uniformly convex norm. Trees in a Banach space are a special instance of vector-valued [[Martingale (probability theory)|martingales]]. Adding techniques from scalar martingale theory, [[Gilles Pisier|Pisier]] improved Enflo's result by showing<ref>Pisier, Gilles (1975), "Martingales with values in uniformly convex spaces", Israel J. Math. '''20''':326–350.</ref>
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| that a super-reflexive space ''X'' admits an equivalent uniformly convex norm for which the modulus of convexity satisfies, for some constant {{nowrap|''c'' > 0}} and some real number {{nowrap|''q'' ≥ 2}},
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| :<math> \delta_X(t) \ge c \, t^q, \quad t \in [0, 2].</math>
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| == Reflexive locally convex spaces ==
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| The notion of reflexive Banach space can be generalized to [[topological vector space]]s in the following way.
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| Let <math>X</math> be a topological vector space over a number field <math>\mathbb F</math> (<math>\mathbb F=</math>[[real number|<math>\mathbb R</math>]] or <math>\mathbb F=</math>[[complex number|<math>\mathbb C</math>]]). Consider its [[Strong topology (polar topology)|strong dual space]] <math>X'_\beta</math>, which consists of all [[continuous function|continuous]] [[linear functional]]s <math>f:X\to {\mathbb F}</math> and is equipped with the [[Strong topology (polar topology)|strong topology]] <math>\beta(X',X)</math>, ''i.e.'', the topology of uniform convergence on bounded subsets in <math>X</math>. The space <math>X'_\beta</math> is a topological vector space (to be more precise, a locally convex space), so one can consider its strong dual space <math>(X'_\beta)'_\beta</math>, which is called the '''strong bidual space''' for <math>X</math>. It consists of all
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| [[continuous function|continuous]] [[linear functional]]s <math>h:X'_\beta\to {\mathbb F}</math> and is equipped with the [[Strong topology (polar topology)|strong topology]] <math>\beta((X'_\beta)',X'_\beta)</math>. Each vector <math>x\in X</math> generates a map <math>J(x):X'_\beta\to{\mathbb F}</math> by the following formula:
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| :<math>
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| J(x)(f)=f(x),\qquad f\in X'.
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| </math>
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| This is a continuous linear functional on <math>X'_\beta</math>, ''i.e.'', <math>J(x)\in (X'_\beta)'_\beta</math>. One obtains a map called '''evaluation map''': | |
| :<math>
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| J:X\to (X'_\beta)'_\beta.
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| </math>
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| This map is linear. If <math>X</math> is locally convex, from the [[Hahn-Banach theorem]] it follows that <math>J</math> is injective and open (''i.e.'', for each neighbourhood of zero <math>U</math> in <math>X</math> there is a neighbourhood of zero <math>V</math> in <math>(X'_\beta)'_\beta</math> such that <math>J(U)\supseteq V\cap J(X)</math>). But it can be non-surjective and/or discontinuous.
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| A locally convex space <math>X</math> is called
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| : - '''semi-reflexive''' if the evaluation map <math>J:X\to (X'_\beta)'_\beta</math> is surjective,
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| : - '''reflexive''' if the evaluation map <math>J:X\to (X'_\beta)'_\beta</math> is surjective and continuous (in this case <math>J</math> is an isomorphism of topological vector spaces).
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| '''Theorem.''' ''A locally convex Hausdorff space <math>X</math> is semi-reflexive if and only <math>X</math> with the <math>\sigma(X, X^*)</math>-topology has the Heine-Broel property (i.e. weakly closed and bounded subsets of <math>X</math> are weakly <math>X</math>).''
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| '''Theorem.'''<ref>{{harvtxt|Schaefer|1966|loc=5.6, 5.5}}</ref> ''A locally convex space <math>X</math> is reflexive if and only if it is semi-reflexive and [[Barrelled space|barreled]].''
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| '''Theorem.''' ''The strong dual of a semireflexive space is barrelled.''
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| === Examples ===
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| 1) Every finite-dimensional Hausdorff [[topological vector space]] is reflexive, because ''J'' is bijective by linear algebra, and because there is a unique Hausdorff vector space topology on a finite dimensional vector space.
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| 2) A normed space <math>X</math> 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 <math>X</math> its dual normed space <math>X'</math> coincides as a topological vector space with the strong dual space <math>X'_\beta</math>. As a corollary, the evaluation map <math>J:X\to X''</math> coincides with the evaluation map <math>J:X\to (X'_\beta)'_\beta</math>, and the following conditions become equivalent:
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| :(i) <math>X</math> is a reflexive normed space (i.e. <math>J:X\to X''</math> is an isomorphism of normed spaces),
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| :(ii) <math>X</math> is a reflexive locally convex space (i.e. <math>J:X\to (X'_\beta)'_\beta</math> is an isomorphism of topological vector spaces),
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| :(iii) <math>X</math> is a semi-reflexive locally convex space (i.e. <math>J:X\to (X'_\beta)'_\beta</math> is surjective).
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| 3) A (somewhat artificial) example of a semi-reflexive space that is not reflexive is obtained as follows: let ''Y'' be an infinite dimensional reflexive Banach space, and let ''X'' be the topological vector space {{nowrap|(''Y'', ''σ''(''Y'', ''Y'' ′))}}, that is, the vector space ''Y'' equipped with the weak topology. Then the continuous dual of ''X'' and {{nowrap|''Y'' ′}} are the same set of functionals, and bounded subsets of ''X'' (''i.e.'', weakly bounded subsets of ''Y'') are norm-bounded, hence the Banach space {{nowrap|''Y'' ′}} is the strong dual of ''X''. Since ''Y'' is reflexive, the continuous dual of {{nowrap|''X'' ′ {{=}} ''Y'' ′}} is equal to the image ''J''(''X'') of ''X'' under the canonical embedding ''J'', but the topology on ''X'' (the weak topology of ''Y'') is not the strong topology {{nowrap|''β''(''X'', ''X'' ′)}}, that is equal to the norm topology of ''Y''.
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| 4) [[Montel space]]s are reflexive locally convex topological vector spaces. In particular, the following functional spaces frequently used in functional analysis are reflexive locally convex spaces:<ref>{{harvtxt|Edwards|1965|loc=8.4.7}}.</ref>
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| * the space <math>C^\infty(M)</math> of smooth functions on arbitrary (real) smooth manifold <math>M</math>, and its strong dual space <math>(C^\infty)'(M)</math> of distributions with compact support on <math>M</math>,
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| * the space <math>{\mathcal D}(M)</math> of smooth functions with compact support on arbitrary (real) smooth manifold <math>M</math>, and its strong dual space <math>{\mathcal D}'(M)</math> of distributions on <math>M</math>,
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| * the space <math>{\mathcal O}(M)</math> of holomorphic functions on arbitrary complex manifold <math>M</math>, and its strong dual space <math>{\mathcal O}'(M)</math> of analytic functionals on <math>M</math>,
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| * the [[Schwartz space]] <math>{\mathcal S}({\mathbb R}^n)</math> on <math>{\mathbb R}^n</math>, and its strong dual space <math>{\mathcal S}'({\mathbb R}^n)</math> of tempered distributions on <math>{\mathbb R}^n</math>.
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| ==Stereotype spaces and other versions of reflexivity==
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| Among all locally convex spaces (even among all Banach spaces) used in [[functional analysis]] the class of reflexive spaces is too narrow to represent a self-sufficient category in any sense. On the other hand, the idea of duality reflected in this notion is so natural that it gives rise to intuitive expectations that appropriate changes in the definition of reflexivity can lead to another notion, more convenient for some goals of mathematics. One of such goals is the idea of approaching analysis to the other parts of mathematics, like [[algebra]] and [[geometry]], by reformulating its results in the purely algebraic language of [[category theory]].
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| This program is being worked out in the theory of [[stereotype space]]s, which are defined as [[topological vector space]]s satisfying a similar condition of reflexivity, but with the topology of uniform convergence on [[totally bounded set|totally bounded]] subsets (instead of [[bounded set|bounded]] subsets) in the definition of dual space X’. More precisely, a topological vector space <math>X</math> is called '''stereotype''' if the evaluation map into the '''stereotype second dual space''' | |
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| :<math> J:X\to X^{\star\star},\quad J(x)(f)=f(x),\quad x\in X,\quad f\in X^\star </math>
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| is an isomorphism of topological vector spaces. Here the ''stereotype dual space'' <math>X^\star</math> is defined as the space of continuous linear functionls <math>X'</math> endowed with the topology of uniform convergence [[totally bounded set]]s in <math>X</math> (and the ''stereotype second dual space'' <math> X^{\star\star}</math> is the space dual to <math> X^{\star}</math> in the same sense).
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| In contrast to the classical reflexive spaces the class '''Ste''' of stereotype spaces is very wide (it contains, in particular, all [[Fréchet space]]s and thus, all [[Banach space]]s), 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.
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| Similarly, one can replace the class of bounded (and totally bounded) subsets in X in the definition of dual space X’, by other classes of subsets, for example, by the class of compact subsets in X -- the spaces defined by the corresponding reflexivity condition are called ''reflective'',<ref>{{cite journal|last=Garibay Bonales|first=F.|coauthors=Trigos-Arrieta, F.J., Vera Mendoza, R.|title=A characterization of Pontryagin-van Kampen duality for locally convex spaces|journal=Topology and its Applications|year=2002|volume=121|pages=75–89}}</ref><ref>{{cite journal|last=Akbarov|first=S. S.|coauthors=Shavgulidze, E. T. |title=On two classes of spaces reflexive in the sense of Pontryagin|journal=Mat. Sbornik|year=2003|volume=194|issue=10|pages=3–26}}</ref> 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'''.
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| ==See also==
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| * A generalization which has some of the properties of reflexive spaces and includes many spaces of practical importance is the concept of [[Grothendieck space]].
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| * [[Reflexive operator algebra]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * [[J.B. Conway]], ''A Course in Functional Analysis'', Springer, 1985.
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| *{{citation
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| | title = Some self-dual properties of normed linear spaces.
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| Symposium on Infinite-Dimensional Topology (Louisiana State Univ., Baton Rouge, La., 1967)
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| | pages = 159–175
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| | series = Ann. of Math. Studies
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| | publisher = Princeton Univ. Press
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| | location = Princeton, N.J.
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| | year = 1972
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| }}.
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| *{{citation
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| | last = Megginson | first = Robert E.
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| | title = An introduction to Banach space theory
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| | series = Graduate Texts in Mathematics
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| | volume = 183
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| | publisher = Springer-Verlag
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| | location = New York
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| | year = 1998
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| | pages = xx+596
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| | isbn = 0-387-98431-3
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| }}.
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| * {{citation
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| | last = Schaefer
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| | first = Helmuth H.
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| <!-- | authorlink = Helmuth Schaefer -->
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| | year = 1966
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| | title = Topological vector spaces
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| | series=
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| | volume=
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| | publisher = The MacMillan Company
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| | location = New York
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| | isbn = 0-387-98726-6
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| }}
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| * {{cite book
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| | last2 = Fomin
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| | first2 = S.V.
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| | year = 1957
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| | title = Elements of the Theory of Functions and Functional Analysis, Volume 1: Metric and Normed Spaces
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| | publisher = Graylock Press
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| | location = Rochester
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| | isbn =
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| }}
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| * {{citation|authorlink=Walter Rudin|first=Walter|last=Rudin|title=Functional analysis|publisher=McGraw-Hill Science|year=1991|isbn=978-0-07-054236-5}}
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| * {{citation
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| | last = Edwards
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| | first = R.E.
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| | year = 1965
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| | title = Functional analysis. Theory and applications
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| | publisher = Holt, Rinehart and Winston
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| | location = New York
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| | isbn =
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| }}
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| * {{cite book | last = Tr\`{e}ves | first = Fran\c{c}ois | title=Topological Vector Spaces, Distributions and Kernels | publisher=[[Academic Press, Inc.]] | year=1995 | isbn=0-486-45352-9 | pages=136-149, 195-201, 240-252, 335-390, 420-433 }}
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| {{Functional Analysis}}
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| [[Category:Banach spaces]]
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| [[Category:Duality theories]]
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