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 $k$,

$$\phi : L \times M \rightarrow k.$$

I.e. $\phi ( a _ {1} l _ {1} + a _ {2} l _ {2} , m)= a _ {1} \phi ( l _ {1} , m)+ a _ {2} \phi ( l _ {2} , m)$, $\phi ( l, b _ {1} m _ {1} + b _ {2} m _ {2} ) = b _ {1} \phi ( l, m _ {1} )+ b _ {2} \phi ( l, m _ {2} )$; $\phi ( l, m)= 0$ for all $m \in M$ implies $l= 0$; $\phi ( l, m)= 0$ 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 $\psi _ {m} : L \rightarrow k$, $\psi _ {m} ( l) = \phi ( l, m)$, 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 $\rho _ {A} ( x) = \sup _ {m \in A } | \phi ( m, x) |$( 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.

References

 [a1] G. Köthe, "Topological vector spaces" , 1 , Springer (1969)
How to Cite This Entry:
Strong topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Strong_topology&oldid=48878