# 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.

• 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 $$$$.

• 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 $$x\mapsto $$ 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 $$F^*$$
• 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. – yada Nov 4 '19 at 14:38
• 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. – yada Nov 4 '19 at 14:41

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 $$M\subseteq E'$$.