functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...

, the Dixmier–Ng theorem is a characterization of when a

normed space The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898. The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war p ...

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 Jacques Dixmier (born 24 May 1924) is a French mathematician. He worked on operator algebras, especially C*-algebras, and wrote several of the standard reference books on them, and introduced the Dixmier trace and the Dixmier mapping. Biogra ...

. : Dixmier-Ng theorem. Let

X

be a normed space. The following are equivalent: # There exists a Hausdorff locally convex topology

\tau

X

so that the

closed unit ball In mathematics, a unit sphere is a sphere of unit radius: the set of points at Euclidean distance 1 from some center point in three-dimensional space. More generally, the ''unit -sphere'' is an -sphere of unit radius in -dimensional Euclidean ...

\mathbf_X

, of

X

\tau

-compact. # There exists a Banach space

Y

so that

X

is isometrically isomorphic to the dual of

Y

. That 2. implies 1. is an application of the

Banach–Alaoglu theorem In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common pro ...

, setting

\tau

to the

Weak-* topology In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...

. That 1. implies 2. is an application of the

Bipolar theorem In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (that is, the polar of the polar) of a set. In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions for a cone ...

Applications

Let

M

be a pointed

metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

with distinguished point denoted

0_M

. The Dixmier-Ng Theorem is applied to show that the Lipschitz space

\text_0(M)

of all real-valued

Lipschitz function In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there e ...

s from

M

\mathbb

that vanish at

0_M

(endowed with the

Lipschitz constant In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exis ...

as norm) is a dual Banach space.

References

{{reflist Theorems in functional analysis