# Dual space (functional analysis)

In mathematics, particularly in the branch of functional analysis, a dual space refers to the space of all continuous linear functionals on a real or complex Banach space. The dual space of a Banach space is again a Banach space when it is endowed with 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 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 $X''$ corresponds to some element of $X$ as described in the next section).

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 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.