Reflexive space

In the area of mathematics known as functional analysis, a reflexive space is a Banach space (or more generally a locally convex topological vector space) that coincides with the 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.Theorem.The Banach space X is super-reflexive if and only if for every 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). In the area of mathematics known as functional analysis, a reflexive space is a Banach space (or more generally a locally convex topological vector space) that coincides with the 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. Suppose X {displaystyle X} is a normed vector space over the number field F = R {displaystyle mathbb {F} =mathbb {R} } or F = C {displaystyle mathbb {F} =mathbb {C} } (the real or complex numbers), with a norm ‖ ⋅ ‖ {displaystyle |cdot |} . Consider its dual normed space X ′ {displaystyle X'} , that consists of all continuous linear functionals f : X → F {displaystyle f:X o {mathbb {F} }} and is equipped with the dual norm ‖ ⋅ ‖ ′ {displaystyle |cdot |'} defined by The dual X ′ {displaystyle X'} is a normed space (a Banach space to be precise), and its dual normed space X ″ = ( X ′ ) ′ {displaystyle X''=(X')'} is called bidual space for X {displaystyle X} . The bidual consists of all continuous linear functionals h : X ′ → F {displaystyle h:X' o {mathbb {F} }} and is equipped with the norm ‖ ⋅ ‖ ″ {displaystyle |cdot |''} dual to ‖ ⋅ ‖ ′ {displaystyle |cdot |'} . Each vector x ∈ X {displaystyle xin X} generates a scalar function J ( x ) : X ′ → F {displaystyle J(x):X' o {mathbb {F} }} by the formula: and J ( x ) {displaystyle J(x)} is a continuous linear functional on X ′ {displaystyle X'} , i.e., J ( x ) ∈ X ″ {displaystyle J(x)in X''} . One obtains in this way a map called evaluation map, that is linear. It follows from the Hahn–Banach theorem that J {displaystyle J} is injective and preserves norms: i.e., J {displaystyle J} maps X {displaystyle X} isometrically onto its image J ( X ) {displaystyle J(X)} in X ″ {displaystyle X''} . Furthermore, the image J ( X ) {displaystyle J(X)} is closed in X ″ {displaystyle X''} , but it need not be equal to X ″ {displaystyle X''} . A normed space X {displaystyle X} is called reflexive if it satisfies the following equivalent conditions: A reflexive space X {displaystyle X} is a Banach space, since X {displaystyle X} is then isometric to the Banach space X ″ {displaystyle X''} . A Banach space X is reflexive if it is linearly isometric to its bidual under this canonical embedding J. 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 J has codimension one in its bidual.A Banach space X is called quasi-reflexive (of order d) if the quotient X ′′ / J(X) has finite dimension d.