From Wikipedia, the free encyclopedia
In
functional analysis, the Dixmier–Ng theorem is a characterization of when a
normed space is in fact a
dual Banach space. It was proven by Kung-fu Ng, who called it a variant of a theorem proven earlier by
Jacques Dixmier.
[1]
[2]
- Dixmier-Ng theorem.
[1] Let
be a normed space. The following are equivalent:
- There exists a
Hausdorff
locally convex topology
on
so that the
closed unit ball,
, of
is
-compact.
- There exists a Banach space
so that
is isometrically isomorphic to the dual of
.
That 2. implies 1. is an application of the
Banach–Alaoglu theorem, setting
to the
Weak-* topology. That 1. implies 2. is an application of the
Bipolar theorem.
Applications
Let
be a
pointed
metric space with distinguished point denoted
. The Dixmier-Ng Theorem is applied to show that the
Lipschitz space
of all real-valued
Lipschitz functions from
to
that vanish at
(endowed with the
Lipschitz constant as norm) is a dual Banach space.
[3]
References
- ^
a
b Ng, Kung-fu (December 1971), "On a theorem of Dixmier", Mathematica Scandinavica, 29: 279–280,
doi:
10.7146/math.scand.a-11054
-
^
Dixmier, J. (December 1948), "Sur un théorème de Banach", Duke Mathematical Journal, 15 (4): 1057–1071,
doi:
10.1215/s0012-7094-48-01595-6
-
^ Godefroy, G.; Kalton, N. J. (2003), "Lipschitz-free Banach spaces", Studia Mathematica, 159 (1): 121–141,
doi:
10.4064/sm159-1-6