Distinguished space

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 (i.e. the strong dual space of their strong dual space) are contained in the weak-* closure of some bounded subset of the bidual. 
Definition.
Suppose that formula_1 is a locally convex space and let formula_2 and formula_3 denote the strong dual of formula_1 (i.e. the continuous dual space of formula_1 endowed with the strong dual topology). 
Let formula_6 denote the continuous dual space of formula_3 and let formula_8 denote the strong dual of formula_9 
Let formula_10 denote formula_6 endowed with the weak-* topology induced by formula_12 where this topology is denoted by formula_13 (that is, the topology of pointwise convergence on formula_2). 
We say that a subset formula_15 of formula_6 is formula_13-bounded if it is a bounded subset of formula_10 and we call the closure of formula_15 in the TVS formula_10 the formula_13-closure of formula_15. 
If formula_23 is a subset of formula_1 then the polar of formula_23 is formula_26 
A Hausdorff locally convex TVS formula_1 is called a distinguished space if it satisfies any of the following equivalent conditions: 
If in addition formula_1 is a metrizable locally convex topological vector space then this list may be extended to include: 
Sufficient conditions.
All normed spaces and semi-reflexive spaces are distinguished spaces. 
LF spaces are distinguished spaces. 
The strong dual space formula_29 of a Fréchet space formula_1 is distinguished if and only if formula_1 is quasibarrelled. 
Properties.
Every locally convex distinguished space is an H-space. 
Examples.
There exist distinguished Banach spaces spaces that are not semi-reflexive. 
The strong dual of a distinguished Banach space is not necessarily separable; formula_32 is such a space. 
The strong dual space of a distinguished Fréchet space is not necessarily metrizable.
There exists a distinguished semi-reflexive non-reflexive -quasibarrelled Mackey space formula_1 whose strong dual is a non-reflexive Banach space. 
There exist H-spaces that are not distinguished spaces. 