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Dual space

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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 $\scriptstyle x \,\in\, X$ .