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

The dual space $$\scriptstyle X'$$ is again a Banach space when it is endowed with the operator norm. Here the operator norm $$\scriptstyle \|f\|$$ of an element $$\scriptstyle f \,\in\, X'$$ is defined as:
 * $$\|f\|=\mathop{\sup}_{x \in X,\,\|x\|_X=1} |f(x)|,$$

where $$\scriptstyle \|\cdot\|_X$$ 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 $$\scriptstyle X''$$. There are special Banach spaces X where one has that $$\scriptstyle X$$ coincides with X (i.e., $$\scriptstyle X\,=\, X$$), in which case one says that X is a reflexive Banach space (to be more precise, $$\scriptstyle X=X$$ here means that every element of $$\scriptstyle X$$ corresponds to some element of $$\scriptstyle X$$ 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 $$\scriptstyle \langle x,x' \rangle $$ between any element $$\scriptstyle x \,\in\, X$$ and any element $$\scriptstyle x' \,\in\, X'$$ defined by
 * $$ \langle x,x' \rangle =x'(x).$$

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

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

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