# Dual space  Main Article Discussion Related Articles  [?] Bibliography  [?] External Links  [?] Citable Version  [?] This editable Main Article is under development and subject to a disclaimer. [edit intro]

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.