Strong topology

A dual pair of vector spaces $(L,M)$ over a field $k$ is a pair of vector spaces $L$, $M$ together with a non-degenerate bilinear form over $k$,

$$\varphi :L\times M\to k.$$

I.e. $\varphi (a_1{l}_1+a_2{l}_2,m)=a_1\varphi (l_1,m)+a_2\varphi (l_2,m)$, $\varphi (l,b_1{m}_1+b_2{m}_2)=b_1\varphi (l,m_1)+b_2\varphi (l,m_2)$; $\varphi (l,m)=0$ for all $m\in M$ implies $l=0$; $\varphi (l,m)=0$ for all $l\in L$ implies $m=0$.

The weak topology on $L$ defined by the dual pair $(L,M)$( given a topology on $k$) is the weakest topology such that all the functionals ${\psi }_m:L\to k$, ${\psi }_m(l)=\varphi (l,m)$, are continuous. More precisely, if $k=\mathbf{R}$ or $\mathbf{C}$ with the usual topology, this defines the weak topology on $L$( and $M$). If $k$ is an arbitrary field with the discrete topology, this defines the so-called linear weak topology.

Let $\mathfrak{M}$ be a collection of bounded subsets of $L$( for the weak topology, i.e. every $A\in \mathfrak{M}$ is weakly bounded, meaning that for every open subset $U$ of $0$ in the weak topology on $L$ there is a $\rho >0$ such that $\rho A\subset U$). The topology ${\tau }_{\mathfrak{M}}$ on $M$ is defined by the system of semi-norms $\{{\rho }_A\}$, $A\in \mathfrak{M}$, where ${\rho }_A(x)=\underset{m\in A}{sup}|\varphi (m,x)|$( cf. Semi-norm). This topology is locally convex if and only if $\cup \mathfrak{M}$ is a total set, i.e. it generates (in $L$ as a vector space) all of $L$. The topology ${\tau }_{\mathfrak{M}}$ is called the topology of uniform convergence on the sets of $\mathfrak{M}$.

The finest topology on $M$ which can be defined in terms of the dual pairs $(L,M)$ is the topology of uniform convergence on weakly bounded subsets of $L$. This is the topology ${\tau }_{\mathfrak{M}}$ where $\mathfrak{M}$ is the collection of all weakly bounded subsets of $L$, and it is called the strong topology on $M$, for brevity.

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