# Dual space

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

- ↑ 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