Two approaches to duality in TVS: dual pairing versus "canonical" dual. Are they equivalent? I have been reading about the basics of duality in Topological Vector Spaces and I have met two different approaches.  
Treves


*

*starts from a TVS $(E,\iota)$, which determines the set of continuous
linear forms on $E$: $E'_\iota$. Then he describes a technique to
topologize $E'_\iota$ through polar topologies on $E'_\iota$.

*He then describes an analogous way to topologise the set of
continuous linear maps $\phi:(E,\tau)\rightarrow (F,\sigma)$ where $(E,\tau)$
and $(F,\sigma)$ are two TVS, let us call this set $L(E,F)$. This is
evidently a generalization of the previous case, if one considers 
$\mathbb{R}$ or $\mathbb{C}$ with the usual topology.  

*Characherizes which topologies $\tau$ on $E$ are such that $E'_\iota=E'_\tau$. Hence, which way of retopologizing a TVS $E$ preserve its topological dual. He calls these topologies compatible with a duality and proves that any topology on a locally convex TVS $E$ can be seen as such: it is the topology of uniform conergence on equicontinuous subsets of $E'$. 

*Moreover he shows that $\sigma(E,E')$, the topology of convergence on finite sets of $E'$ is the coarsest to achieve the result and moreover he shows we can always embed $E$ in $(E'_\sigma)'$.

*Proves the Mackey theorem stating that topologies compatible with a duality share the same bounded sets, and treat reflexivity.


Schechter


*

*Start from an abstract dual pair $\langle E,F\rangle$: this is a completely algebraic construction. $E$ and $F$ are two vector spaces and $\langle,\rangle$ is a (separating) dual form. 

*Describe polar topologies on $E$ or on $F$ in an analogous way as Treves. Now, however this is done through $\langle,\rangle$ and not the canonical evaluation of elements of the dual $E'$. 

*Describe how the topology $\sigma(F,G)$ is the weakest making all the evaluations of elements of $G$ aigainst element of $F$ continuous, and gives both the result about the topology of equicontinuous convergence and the embedding in the double dual.
Now I ask: 


*

*Are these two approaches completely equivalent?


It seems that, obviously, the Schechter's approach includes Treves by considering the canonical evaluation of a functional of $E'$ angainst an element of $E$.  Anyhow, I personally find Treves' discussion neater, and moreover all examples I encountered so far are of the type $<E,E'>$.  


*

*But can this appproach be shown equivalent to the former?

*If we start from a dual pair $\langle F,G\rangle$ can we always
recover all informations about it by considering some topology on $F$
and $\langle F,F'\rangle$?

*If not which are the differences between the two?


What I can notice, at the algebraic level is the following rather straghtforwad fact


*

*A vector space $E$ and its algebraic dual $E^*$ form a dual pair
$\langle E,E^*\rangle$ where the pairing bilinear map is given by the
evaluation of linear functionals: $(x\in E, y\in E^*)\mapsto y(x)\in
   \mathbb{R}$.

*Conversely, given an arbitrary separating dual pair $\langle
   E,F\rangle$, we can notice that the bilinear form $\langle,\rangle$,
a fixed $y\in F$ induces a linear map $x\mapsto <x,y>$ which is
linear and injective. Hence it is an element of $E^*$ then $F$ can
always  be indentified with a subspace of $E^*$.
Or viceversa, with $E$ and $F^*$
 A: Given a dual pair $\langle E,F\rangle$ you can endow $E$ with the weak topology $\sigma(E,F)$ defined by the seminorms $$p_J(x)=\max\{|\langle x,y\rangle|: y\in J\}$$ for finite subsets $J$ of $F$. Then the linear map $\Phi:F\to (E,\sigma(E,F))'$, $y\mapsto \langle \cdot,y\rangle$ is a bijection (injective because the duality is separating, for the surjectivity you need a standard lemma: For all linear functionals $\phi,\phi_1,\ldots,\phi_n$ on $E$ such that the intersection of the kernels of $\phi_i$ is contained in the kernel of $\phi$ you get $\phi$ as a linear combination of the $\phi_i$).
The advantage of general dual pairs is the symmetry. For example, if you have proved the theorem of bipolars that $B^{\circ\bullet}$ is the closed absolutely convex hull of $B\subseteq E$ (where $B^\circ$ is the polar in $E'$ and $M^\bullet$ is the polar in $E$) you get without further ado also the dual version that $M^{\bullet\circ}$ is the $\sigma(E',E)$-closed absolutely convex hull of $M\subseteq E'$.
