# Dual space

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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