What is a Functional? Definition & Uses

[observer1]

In the calculus of variations, the integral itself is a "functional." It depends on the form of the function of the Lagrangian: q and q-dot

But I have seen this word "functional" used elsewhere in different contexts.

I have seen: "A functional is a real valued function on a vector space."

I have seen: "A function is a function of functions."

I understand how these are all used, but I am still bereft of a "precise" and "consistent" definition of this word.
Can someone provide it? (Perhaps the key is that there is no "consistent" definition?)

 

[fresh_42]

Functions can also build a vector space and in a very large number of cases they actually do: continuous functions, smooth functions, linear functions ... Thus there isn't necessarily a contradiction between the two definitions. Wikipedia has the first one but says "commonly".
This means that it depends a little bit on the author, resp. the context. Would you ask what a number is, too? Integers, reals, or quaternions? Why are they sometimes called scalars? Some terms don't have a rigor definition although "A functional is a mapping from a vector space to the underlying field." is rather close.

 

[observer1]

and thank you!

 

[Stephen Tashi]

Mathematical terminology isn't necessarily consistent.

However, in the case of "functional", the notion that a "functional" is a function who domain is a set of functions and whose range is a set of numbers is consistent with the notion that a functional is a function who domain is a set of vectors and whose range is a set of numbers. The two ideas are consistent because "functions" can be viewed as an example of "vectors".

Consider real valued functions of one real variable. You can add vectors. You can add functions. You can multiply vectors by a scalar to get another vector. You can multiply functions by a scalar to get another function. You can compute the inner product of two vectors f and g. You can compute $\int f(x) g(x) dx$ which acts like an inner product between functions - if the integral converges.

If you provide a specific formula for a functional $L$, like $L(f(x)) = \int_{-\infty}^{\infty} f(x) e^{-x} dx$ this is just an example of the general notion that the functional $L$ is a function that maps functions to numbers.

 

[chiro]

Hey observer1.

I'd try and look at learning the Fourier Series decomposition (not the same as a Laplace transform or other similar ones) where you can get a vector of coefficients for each harmonic and if you represent that as an infinite vector then the vector itself represents your functional in that space.

 

[Ssnow]

If in the usual calculus you have $f$ the function and $x$ the variable, a functional is the analogue where now the function is $F$ and the variable is itself a function $f$ so the expression '' function of functions''. As in the case of the classical functions you must specify the domain of $f$ that can be continuous, smooth, ... so it works in similar way
A simple example of functional is the definite integral $\int_{a}^{b} \cdot dx$ and the valued is in a field (ex. $\mathbb{R}$). A functional can be linear $F(f+g)=F(f)+F(g)$ as the integral and as said before considering function as vectors you can consider a inner product and so on ...

Note: a general functional can be also nonlinear and that you can iterate the process considering functional of functionals $F(F(\cdots ))$, but these are secondary considerations...