Dual space (functional analysis)

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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 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'.


[edit] 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.

[edit] 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.

[edit] 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[1].

[edit] References

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

[edit] Further reading

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

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