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'.
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.
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.
- R. T. Rockafellar, Conjugate Duality and Optimization, CBMS Reg. Conf. Ser. Appl. Math. 16, SIAM, Philadelphia, 1974
K. Yosida, Functional Analysis (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980