PlanetMath: functional

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functional

(Definition)

Definition

A functional $T$ is a function mapping a function space (often a vector space) $V$ in to a field of scalars $K$ , typically taken to be $\R$ or $\mathbb{C}$ .

Discussion

Examples of functionals include the integral and entropy. A functional $f$ is often indicated by the use of square brackets, $T[x]$ rather than $T(x)$ .

The linear functionals are those functionals $T$ that satisfy

$T(x+y)=T(x)+T(y)$
$T(cx)=cT(x)$

for any $c\in K$ , $x,y\in V$ .

"functional" is owned by mathcam. [ full author list (2) | owner history (1) ]

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Also defines:

linear functional

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Cross-references: entropy, integral, functionals, scalars, field, vector space, function space, mapping, function
There are 47 references to this entry.

This is version 6 of functional, born on 2002-02-26, modified 2005-03-18.
Object id is 2722, canonical name is Functional.
Accessed 11710 times total.

Classification:

AMS MSC:

03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )

Pending Errata and Addenda

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