• Source: Semi-reflexive space

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 (which is the strong dual of X) is bijective.
If this map is also an isomorphism of TVSs then it is called reflexive.
Semi-reflexive spaces play an important role in the general theory of locally convex TVSs.
Since a normable TVS is semi-reflexive if and only if it is reflexive, the concept of semi-reflexivity is primarily used with TVSs that are not normable.




Definition and notation






= Brief definition

=
Suppose that X is a topological vector space (TVS) over the field




F



{\displaystyle \mathbb {F} }

(which is either the real or complex numbers) whose continuous dual space,




X

′




{\displaystyle X^{\prime }}

, separates points on X (i.e. for any



x
∈
X


{\displaystyle x\in X}

there exists some




x

′


∈

X

′




{\displaystyle x^{\prime }\in X^{\prime }}

such that




x

′


(
x
)
≠
0


{\displaystyle x^{\prime }(x)\neq 0}

).
Let




X

b


′




{\displaystyle X_{b}^{\prime }}

and




X

β


′




{\displaystyle X_{\beta }^{\prime }}

both denote the strong dual of X, which is the vector space




X

′




{\displaystyle X^{\prime }}

of continuous linear functionals on X endowed with the topology of uniform convergence on bounded subsets of X;
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 X is a normed space, then the strong dual of X is the continuous dual space




X

′




{\displaystyle X^{\prime }}

with its usual norm topology.
The bidual of X, denoted by




X

′
′




{\displaystyle X^{\prime \prime }}

, is the strong dual of




X

b


′




{\displaystyle X_{b}^{\prime }}

; that is, it is the space





(

X

b


′


)


b


′




{\displaystyle \left(X_{b}^{\prime }\right)_{b}^{\prime }}

.
For any



x
∈
X
,


{\displaystyle x\in X,}

let




J

x


:

X

′


→

F



{\displaystyle J_{x}:X^{\prime }\to \mathbb {F} }

be defined by




J

x



(

x

′


)

=

x

′


(
x
)


{\displaystyle J_{x}\left(x^{\prime }\right)=x^{\prime }(x)}

, where




J

x




{\displaystyle J_{x}}

is called the evaluation map at x;
since




J

x


:

X

b


′


→

F



{\displaystyle J_{x}:X_{b}^{\prime }\to \mathbb {F} }

is necessarily continuous, it follows that




J

x


∈


(

X

b


′


)


′




{\displaystyle J_{x}\in \left(X_{b}^{\prime }\right)^{\prime }}

.
Since




X

′




{\displaystyle X^{\prime }}

separates points on X, the map



J
:
X
→


(

X

b


′


)


′




{\displaystyle J:X\to \left(X_{b}^{\prime }\right)^{\prime }}

defined by



J
(
x
)
:=

J

x




{\displaystyle J(x):=J_{x}}

is injective where this map is called the evaluation map or the canonical map.
This map was introduced by Hans Hahn in 1927.
We call X semireflexive if



J
:
X
→


(

X

b


′


)


′




{\displaystyle J:X\to \left(X_{b}^{\prime }\right)^{\prime }}

is bijective (or equivalently, surjective) and we call X reflexive if in addition



J
:
X
→

X

′
′


=


(

X

b


′


)


b


′




{\displaystyle J:X\to X^{\prime \prime }=\left(X_{b}^{\prime }\right)_{b}^{\prime }}

is an isomorphism of TVSs.
If X is a normed space then J is a TVS-embedding as well as an isometry onto its range;
furthermore, by Goldstine's theorem (proved in 1938), the range of J is a dense subset of the bidual




(


X

′
′


,
σ

(


X

′
′


,

X

′



)


)



{\displaystyle \left(X^{\prime \prime },\sigma \left(X^{\prime \prime },X^{\prime }\right)\right)}

.
A normable space is reflexive if and only if it is semi-reflexive.
A Banach space is reflexive if and only if its closed unit ball is



σ

(


X

′


,
X

)



{\displaystyle \sigma \left(X^{\prime },X\right)}

-compact.




= Detailed definition

=
Let X be a topological vector space over a number field




F



{\displaystyle \mathbb {F} }

(of real numbers




R



{\displaystyle \mathbb {R} }

or complex numbers




C



{\displaystyle \mathbb {C} }

).
Consider its strong dual space




X

b


′




{\displaystyle X_{b}^{\prime }}

, which consists of all continuous linear functionals



f
:
X
→

F



{\displaystyle f:X\to \mathbb {F} }

and is equipped with the strong topology



b

(


X

′


,
X

)



{\displaystyle b\left(X^{\prime },X\right)}

, that is, the topology of uniform convergence on bounded subsets in X.
The space




X

b


′




{\displaystyle X_{b}^{\prime }}

is a topological vector space (to be more precise, a locally convex space), so one can consider its strong dual space





(

X

b


′


)


b


′




{\displaystyle \left(X_{b}^{\prime }\right)_{b}^{\prime }}

, which is called the strong bidual space for X.
It consists of all
continuous linear functionals



h
:

X

b


′


→


F




{\displaystyle h:X_{b}^{\prime }\to {\mathbb {F} }}

and is equipped with the strong topology



b

(



(

X

b


′


)


′


,

X

b


′



)



{\displaystyle b\left(\left(X_{b}^{\prime }\right)^{\prime },X_{b}^{\prime }\right)}

.
Each vector



x
∈
X


{\displaystyle x\in X}

generates a map



J
(
x
)
:

X

b


′


→

F



{\displaystyle J(x):X_{b}^{\prime }\to \mathbb {F} }

by the following formula:




J
(
x
)
(
f
)
=
f
(
x
)
,

f
∈

X
′

.


{\displaystyle J(x)(f)=f(x),\qquad f\in X'.}


This is a continuous linear functional on




X

b


′




{\displaystyle X_{b}^{\prime }}

, that is,



J
(
x
)
∈


(

X

b


′


)


b


′




{\displaystyle J(x)\in \left(X_{b}^{\prime }\right)_{b}^{\prime }}

.
One obtains a map called the evaluation map or the canonical injection:




J
:
X
→


(

X

b


′


)


b


′


.


{\displaystyle J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }.}


which is a linear map.
If X is locally convex, from the Hahn–Banach theorem it follows that J is injective and open (that is, for each neighbourhood of zero



U


{\displaystyle U}

in X there is a neighbourhood of zero V in





(

X

b


′


)


b


′




{\displaystyle \left(X_{b}^{\prime }\right)_{b}^{\prime }}

such that



J
(
U
)
⊇
V
∩
J
(
X
)


{\displaystyle J(U)\supseteq V\cap J(X)}

).
But it can be non-surjective and/or discontinuous.
A locally convex space



X


{\displaystyle X}

is called semi-reflexive if the evaluation map



J
:
X
→


(

X

b


′


)


b


′




{\displaystyle J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }}

is surjective (hence bijective); it is called reflexive if the evaluation map



J
:
X
→


(

X

b


′


)


b


′




{\displaystyle J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }}

is surjective and continuous, in which case J will be an isomorphism of TVSs).




Characterizations of semi-reflexive spaces


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

X is semireflexive;
the weak topology on X had the Heine-Borel property (that is, for the weak topology



σ

(

X
,

X

′



)



{\displaystyle \sigma \left(X,X^{\prime }\right)}

, every closed and bounded subset of




X

σ




{\displaystyle X_{\sigma }}

is weakly compact).
If linear form on




X

′




{\displaystyle X^{\prime }}

that continuous when




X

′




{\displaystyle X^{\prime }}

has the strong dual topology, then it is continuous when




X

′




{\displaystyle X^{\prime }}

has the weak topology;





X

τ


′




{\displaystyle X_{\tau }^{\prime }}

is barrelled, where the



τ


{\displaystyle \tau }

indicates the Mackey topology on




X

′




{\displaystyle X^{\prime }}

;
X weak the weak topology



σ

(

X
,

X

′



)



{\displaystyle \sigma \left(X,X^{\prime }\right)}

is quasi-complete.




Sufficient conditions


Every semi-Montel space is semi-reflexive and every Montel space is reflexive.




Properties


If



X


{\displaystyle X}

is a Hausdorff locally convex space then the canonical injection from



X


{\displaystyle X}

into its bidual is a topological embedding if and only if



X


{\displaystyle X}

is infrabarrelled.
The strong dual of a semireflexive space is barrelled.
Every semi-reflexive space is quasi-complete.
Every semi-reflexive normed space is a reflexive Banach space.
The strong dual of a semireflexive space is barrelled.




Reflexive spaces



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

X is reflexive;
X is semireflexive and barrelled;
X is barrelled and the weak topology on X had the Heine-Borel property (which means that for the weak topology



σ

(

X
,

X

′



)



{\displaystyle \sigma \left(X,X^{\prime }\right)}

, every closed and bounded subset of




X

σ




{\displaystyle X_{\sigma }}

is weakly compact).
X is semireflexive and quasibarrelled.
If X is a normed space then the following are equivalent:

X is reflexive;
the closed unit ball is compact when X has the weak topology



σ

(

X
,

X

′



)



{\displaystyle \sigma \left(X,X^{\prime }\right)}

.
X is a Banach space and




X

b


′




{\displaystyle X_{b}^{\prime }}

is reflexive.




Examples


Every non-reflexive infinite-dimensional Banach space is a distinguished space that is not semi-reflexive.
If



X


{\displaystyle X}

is a dense proper vector subspace of a reflexive Banach space then



X


{\displaystyle X}

is a normed space that not semi-reflexive but its strong dual space is a reflexive Banach space.
There exists a semi-reflexive countably barrelled space that is not barrelled.




See also


Grothendieck space - A generalization which has some of the properties of reflexive spaces and includes many spaces of practical importance.
Reflexive operator algebra
Reflexive space




Citations






Bibliography


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John B. Conway, A Course in Functional Analysis, Springer, 1985.
James, Robert C. (1972), Some self-dual properties of normed linear spaces. Symposium on Infinite-Dimensional Topology (Louisiana State Univ., Baton Rouge, La., 1967), Ann. of Math. Studies, vol. 69, Princeton, NJ: Princeton Univ. Press, pp. 159–175.
Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
Kolmogorov, A. N.; Fomin, S. V. (1957). Elements of the Theory of Functions and Functional Analysis, Volume 1: Metric and Normed Spaces. Rochester: Graylock Press.
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Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.
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Kata Kunci Pencarian:

• Reflexive space
• Semi-reflexive space
• Strong dual space
• Banach space
• Quasi-complete space
• Infrabarrelled space
• Nuclear space
• Sesquilinear form
• Distinguished space
• Montel space