Editing Reflexive space (section)

== Definition ==

;Definition of the bidual

{{Main|Bidual}}

Suppose that <math>X</math> is a [[topological vector space]] (TVS) over the field <math>\mathbb{F}</math> (which is either the real or complex numbers) whose [[continuous dual space]], <math>X^{\prime},</math> '''separates points''' on <math>X</math> (that is, for any <math>x \in X, x \neq 0</math> there exists some <math>x^{\prime} \in X^{\prime}</math> such that <math>x^{\prime}(x) \neq 0</math>). 
Let <math>X^{\prime}_b</math> (some texts write <math>X^{\prime}_\beta</math>) denote the [[strong dual]] of <math>X,</math> which is the vector space <math>X^{\prime}</math> of continuous linear functionals on <math>X</math> endowed with the [[topology of uniform convergence]] on [[Topology of uniform convergence#Strong dual topology b(X*, X)|bounded subsets]] of <math>X</math>; 
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 <math>X</math> is a normed space, then the strong dual of <math>X</math> is the continuous dual space <math>X^{\prime}</math> with its usual norm topology. 
The '''bidual''' of <math>X,</math> denoted by <math>X^{\prime\prime},</math> is the strong dual of <math>X^{\prime}_b</math>; that is, it is the space <math>\left(X^{\prime}_b\right)^{\prime}_b.</math>{{sfn|Trèves|2006|pp=372-374}} 
If <math>X</math> is a normed space, then <math>X^{\prime\prime}</math> is the continuous dual space of the Banach space <math>X^{\prime}_b</math> with its usual norm topology. 

;Definitions of the evaluation map and reflexive spaces

For any <math>x \in X,</math> let <math>J_x : X^{\prime} \to \mathbb{F}</math> be defined by <math>J_x\left(x^{\prime}\right) = x^{\prime}(x),</math> where <math>J_x</math> is a linear map called the '''evaluation map at <math>x</math>'''; 
since <math>J_x : X^{\prime}_b \to \mathbb{F}</math> is necessarily continuous, it follows that <math>J_x \in \left(X^{\prime}_b\right)^{\prime}.</math> 
Since <math>X^{\prime}</math> separates points on <math>X,</math> the linear map <math>J : X \to \left(X^{\prime}_b\right)^{\prime}</math> defined by <math>J(x) := J_x</math> is injective where this map is called the '''evaluation map''' or the '''canonical map'''. 
Call <math>X</math> '''[[semi-reflexive]]''' if <math>J : X \to \left(X^{\prime}_b\right)^{\prime}</math> is bijective (or equivalently, [[surjective]]) and we call <math>X</math> '''reflexive''' if in addition <math>J : X \to X^{\prime\prime} = \left(X^{\prime}_b\right)^{\prime}_b</math> is an isomorphism of TVSs.{{sfn|Trèves|2006|pp=372-374}}
A [[normable]] space is reflexive if and only if it is semi-reflexive or equivalently, if and only if the evaluation map is surjective.