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.

 

[blue_raver22]

Based on the given information, it appears that $f$ is a function. A function is a mapping between two sets, in this case, from a vector space to a scalar space. A functional, on the other hand, is a mapping from a vector space to its underlying field. In other words, a functional takes in a vector and returns a number, while a function can take in any type of input and return any type of output.

In this case, $f$ is taking in a vector $\textbf{x}$ and returning a scalar, which aligns with the definition of a function. However, it is important to note that the terms function and functional can be used interchangeably in certain contexts, so it ultimately depends on the specific field or context in which this mapping is being discussed.