Strong dual space

In functional analysis, the strong dual of a topological vector space (TVS) X is the continuous dual space of X equipped with the strong topology or the topology of uniform convergence on bounded subsets of X, where this topology is denoted by ${\displaystyle b\left(X^{\prime },X\right)}$ or ${\displaystyle \beta \left(X^{\prime },X\right)}$. The strong dual space plays such an important role in modern functional analysis, that the continuous dual space is usually assumed to have the strong dual topology unless indicated otherwise. To emphasize that the continuous dual space, ${\displaystyle X^{\prime }}$, has the strong dual topology, ${\displaystyle X_{b}^{\prime }}$ or ${\displaystyle X_{\beta }^{\prime }}$ may be written.

Strong dual topology

Definition from a dual system

Let ${\displaystyle (X,Y,\left\langle \cdot ,\cdot \right\rangle )}$ be a dual system of vector spaces over the field ${\displaystyle {\mathbb {F} }}$ of real (${\displaystyle {\mathbb {R} }}$) or complex (${\displaystyle {\mathbb {C} }}$) numbers. Note that neither X nor Y has a topology so we define a subset B of X to be bounded if and only if ${\displaystyle \sup _{x\in B}\left|\left\langle x,y\right\rangle \right|<\infty }$ for all ${\displaystyle y\in Y}$. This is equivalent to the usual notion of bounded subsets when X is given the weak topology induced by Y, which is a Hausdorff locally convex topology. The definition of the strong dual topology now proceeds as in the case of a TVS.

Note that if X is a TVS whose continuous dual space separates point on X, then X is part of a canonical dual system ${\displaystyle \left(X,X^{\prime },\left\langle \cdot ,\cdot \right\rangle \right)}$ where ${\displaystyle \left\langle x,x^{\prime }\right\rangle$ :=x^{\prime }(x)} .

Definition on a TVS

Suppose that X is a topological vector space (TVS) over the field ${\displaystyle {\mathbb {F} }}$ of real (${\displaystyle {\mathbb {R} }}$) or complex (${\displaystyle {\mathbb {C} }}$) numbers. Let ${\displaystyle {\mathcal {B}}}$ be any fundamental system of bounded sets of X (i.e. a set of bounded subsets of X such that every bounded subset of X is a subset of some ${\displaystyle B\in {\mathcal {B}}}$); the set of all bounded subsets of X trivially forms a fundamental system of bounded sets of X. A basis of closed neighborhoods of the origin in ${\displaystyle X^{\prime }}$ is given by the polars:

${\displaystyle B^{\circ }:=\left\{x^{\prime }\in X^{\prime }:\sup _{x\in B}\left|x^{\prime }\left(x\right)\right|\leq 1\right\}}$

as B ranges over ${\displaystyle {\mathcal {B}}}$). This is a locally convex topology that is given by the set of seminorms on ${\displaystyle X^{\prime }}$: ${\displaystyle \left|x^{\prime }\right|_{B}:=\sup _{x\in B}\left|x^{\prime }\left(x\right)\right|}$ as B ranges over ${\displaystyle {\mathcal {B}}}$.

If X is normable then so is ${\displaystyle X_{b}^{\prime }}$ and ${\displaystyle X_{b}^{\prime }}$ will in fact be a Banach space. If X is a normed space with norm ${\displaystyle \|\cdot \|}$ then ${\displaystyle X^{\prime }}$ has a canonical norm (the operator norm) given by ${\displaystyle \left\|x^{\prime }\right\|:=\sup _{\|x\|\leq 1}\left|x^{\prime }(x)\right|}$; the topology that this norm induces on ${\displaystyle X^{\prime }}$ is identical to the strong dual topology.

Properties

Let X be a locally convex TVS.

• A convex, balanced, weakly compact subset of X' is bounded in ${\displaystyle X_{b}^{\prime }}$.[1]
• Every weakly bounded subset of X' is strongly bounded.[2]
• If X is a barreled space then X's topology is identical to the strong dual topology b(X, X') and to the Mackey topology on X.
• If X is a metrizable locally convex space, then the strong dual of X is bornological if and only if it is infrabarreled, if and only if it is barreled.[3]
• If X is Hausdorff locally convex TVS then (X, b(X, X')) is metrizable if and only if there exists a countable set ℬ of bounded subsets of X such that every bounded subset of X is contained in some element of ℬ.[4]

See also

• Dual topology
• Dual system
• List of topologies
• Polar topology – Dual space topology of uniform convergence on some sub-collection of bounded subsets
• Strong topology
• Strong topology (polar topology) – Dual space topology of uniform convergence on bounded subsets
• Topologies on spaces of linear maps

References

1. Schaefer & Wolff 1999, p. 141.
2. Schaefer & Wolff 1999, p. 142.
3. Schaefer & Wolff 1999, p. 153.
4. Narici & Beckenstein 2011, pp. 225-273.

Bibliography

• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
• Wong (1979). Schwartz spaces, nuclear spaces, and tensor products. Berlin New York: Springer-Verlag. ISBN 3-540-09513-6. OCLC 5126158.