Dual space

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 $\scriptstyle X$ 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 topology induced by 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 $\scriptstyle X''$ is in a one-to-one correspondence with an element of $\scriptstyle X$).

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 $\scriptstyle \langle x,x'\rangle$ 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 $X'$ 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.

Further reading

K. Yosida, Functional Analysis (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980