Does a "functional" always takes in a function and spit out a number?

This is what a professor said in class a long time ago but now I am studying Frechet derivative and a claim was made that a linear functional is $Ax: V \to W$. Can $W$ be anything other than a field i.e. a functional space?

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    $\begingroup$ 1) Yes, functional maps into the underlying scalar field. 2) Yes, for example $W$ can be the space of continuous linear functionals on $V$. $\endgroup$ – user251257 Jul 15 '15 at 17:13
  • $\begingroup$ @user251257: But the dual space of $V$ is also a space consisting of functions :) $\endgroup$ – gerw Jul 15 '15 at 17:52
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    $\begingroup$ @gerw functional are functions ... I don't get your point. $\endgroup$ – user251257 Jul 15 '15 at 17:54
  • $\begingroup$ @user251257: I think I misunderstood the last sentence in the question.. Sorry! $\endgroup$ – gerw Jul 15 '15 at 17:59

A functional is a map from a vector space $V$ to its underlying scalar field $F$. So not quite. For example the norm on $\Bbb{R}^3$ is a functional because it maps the vector $(x,y,z)$ to the real number $\sqrt{x^2+y^2+z^2}$.

If you have a function space (a vector space of functions), then a functional is a map from the vector space (functions) to the scalar field (numbers). For example on $L^1(\Bbb{R})$ the integral which maps $f$ to $\int_{\Bbb{R}} fdx$ is a functional.

A functional can map functions to numbers, and this is the context you'll see it mostly. Or other vectors to other scalars. Functionals defined on exotic vector spaces are still functionals.

  • $\begingroup$ Your example is actually badly defined, you should use $L^1$ or a finite measure subset of $\mathbb{R}$ instead. $\endgroup$ – Ian Jul 15 '15 at 19:32

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