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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 $F^*$
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    $\begingroup$ Both approaches are equivalent. The advantage of starting with a dual pair of vector spaces is that you don't need to define any topology in advance. For instance, we can consider the duality between bounded measurable functions $M_b(T, \Sigma)$ and finite signed measures $ca(\Sigma)$ given by integration. If you however equip $M_b(X, \Sigma)$ with its usual supremum norm then its dual is much larger: it is the space $ba(\Sigma)$ of all finitely additive signed measures with bounded variation. So, if you just consider a dual pair, you don't need to care about identifying topological duals. $\endgroup$ – yada Nov 4 '19 at 14:38
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    $\begingroup$ Now, considering the dual pair $\langle M_b, ca \rangle$ you can just speak of the weak topology $\sigma(M_b, ca)$ or the Mackey topology $\mu(M_b, ca)$ (or any other locally convex topology in between) which all are compatible with the dual pair, meaning that the dual of $M_b$ equipped with this topology is precisely what you want, namely the $ca$-space. $\endgroup$ – yada Nov 4 '19 at 14:41
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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 $M\subseteq E'$.

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