# Dual space

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

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

${\displaystyle \|f\|=\mathop {\sup } _{x\in X,\,\|x\|_{X}=1}|f(x)|,}$

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

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 ${\displaystyle \scriptstyle \langle x,x'\rangle }$ between any element ${\displaystyle \scriptstyle x\,\in \,X}$ and any element ${\displaystyle \scriptstyle x'\,\in \,X'}$ defined by

${\displaystyle \langle x,x'\rangle =x'(x).}$

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

${\displaystyle x(x')=\langle x,x'\rangle =x'(x),}$

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

Dual pairings play an important role in many branches of mathematics, for example in the duality theory of convex optimization[1].

## References

1. R. T. Rockafellar, Conjugate Duality and Optimization, CBMS Reg. Conf. Ser. Appl. Math. 16, SIAM, Philadelphia, 1974