Dual space

From Citizendium
Jump to: navigation, search
This article is developing and not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

In mathematics, particularly in the branch of functional analysis, a dual space refers to the space of all continuous linear functionals of a real or complex Banach space. The dual space of a Banach space is again a Banach space when it is endowed with the topology induced by the operator norm. If X is a Banach space then its dual space is often denoted by X'.

Definition

Let X be a Banach space over a field F which is real or complex, then the dual space X' of is the vector space over F of all continuous linear functionals when F is endowed with the standard Euclidean topology.

The dual space is again a Banach space when it is endowed with the topology induced by the operator norm. Here the operator norm of an element is defined as:

where denotes the norm on X.

The bidual space and reflexive Banach spaces

Since X' is also a Banach space, one may define the dual space of the dual, often referred to as a bidual space of X and denoted as . There are special Banach spaces X where one has that coincides with X (i.e., ), in which case one says that X is a reflexive Banach space (to be more precise, here means that every element of is in a one-to-one correspondence with an element of ).

An important class of reflexive Banach spaces are the Hilbert spaces, i.e., every Hilbert space is a reflexive Banach space. This follows from an important result known as the Riesz representation theorem.

Dual pairings

If X is a reflexive Banach space then one may define a bilinear form or pairing between any element and any element defined by

Notice that defines a continuous linear functional on X for each , while defines a continuous linear functional on for each . It is often convenient to also express

i.e., a continuous linear functional f on is identified as for a unique element . For a reflexive Banach space such bilinear pairings determine all continuous linear functionals on X and since it holds that every functional with can be expressed as for some unique element .

Dual pairings play an important role in many branches of mathematics, for example in the duality theory of convex optimization[1].

References

  1. R. T. Rockafellar, Conjugate Duality and Optimization, CBMS Reg. Conf. Ser. Appl. Math. 16, SIAM, Philadelphia, 1974

Further reading

K. Yosida, Functional Analysis (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980