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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 $X$ is the vector space over F of all continuous linear functionals $f:\,X\rightarrow \,F$ when F is endowed with the standard Euclidean topology.

The dual space $X''$ is again a Banach space when it is endowed with the topology induced by the operator norm. Here the operator norm $\|f\|$ of an element $f\,\in \,X'$ is defined as:

$\|f\|=\mathop {\sup } _{x\in X,\,\|x\|_{X}=1}|f(x)|,$ where $\|\cdot \|_{X}$ 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 $X''$ . There are special Banach spaces X where one has that $X''$ coincides with X (i.e., $X''\,=\,X$ ), in which case one says that X is a reflexive Banach space (to be more precise, $X''=X$ here means that every element of $X''$ is in a one-to-one correspondence with an element of $X$ ).

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 $\langle x,x'\rangle$ between any element $x\,\in \,X$ and any element $x'\,\in \,X'$ defined by

$\langle x,x'\rangle =x'(x).$ Notice that $\langle \cdot ,x'\rangle$ defines a continuous linear functional on X for each $x'\,\in \,X'$ , while $\langle x,\cdot \rangle$ defines a continuous linear functional on $X'$ for each $x\,\in \,X$ . It is often convenient to also express

$x(x')=\langle x,x'\rangle =x'(x),$ i.e., a continuous linear functional f on $X'$ is identified as $f(x')\,=\,\langle x,x'\rangle$ for a unique element $x\,\in \,X$ . For a reflexive Banach space such bilinear pairings determine all continuous linear functionals on X and $X'$ since it holds that every functional $x''(x')$ with $x''\,\in \,X''$ can be expressed as $x''(x')\,=\,x'(x)$ for some unique element $x\,\in \,X$ .

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