MHB Is $f$ a function or a functional?

[OhMyMarkov]

Hello everyone!

I'm a bit confused about referring to a mapping as function or functional, for example: $f(x_1, x_2, x_3) = x_1+2x_2 ^2+3x_3 ^3$. $f$ takes vector $\textbf{x}=[x_1 \; x_2 \; x_3]$ and maps it to a scalar. Now, is $f$ a function or a functional?

Thanks!

 

[Ackbach]

It's both, in this case. A functional is a function that maps a vector from a vector space into the field over which the vector space is defined. So all functionals are functions, but not all functions are functionals. Does that help?

 

[OhMyMarkov]

It's both, in this case. A functional is a function that maps a vector from a vector space into the field over which the vector space is defined. So all functionals are functions, but not all functions are functionals. Does that help?

Not really sure unless, say, a counter example would be a continuous transform such as the continuous Fourier transform?

 

[Ackbach]

Not really sure unless, say, a counter example would be a continuous transform such as the continuous Fourier transform?

Sure. The result of a Fourier transform is another function (or vector, depending on your vector space). Another counterexample is any matrix transformation, such as a rotation.