What is the difference between a function and a functional?

[zhermes]

My background is in physics, not pure mathematics, so please try to explain in ways that we lay-people could understand ;)
I'm brushing up on my calculus of variations--specifically Hamilton's principle--in which it is stated that the integrand is a 'functional,' not a 'function.' I've read that a 'functional' is a mapping from a vector space to a scalar, e.g. from a vector space to its underlying field--but I don't quite understand the significance of this. If someone could elaborate on the explanation, or provide a physically-motivated example, that would be very helpful!

 

[arkajad]

Suppose f is a function on R (like f(x)=sin(x)). Fix a point a.
the assign to each function f its value f(a) at a. The map

$$f\mapsto f(a)$$

assigns to each function a number - its value at the point a. It is an example of a functional. Another example. Chose two points a,b. Assign to each function f the number

$$\int_a^b f(x)^2\, dx$$

You have another example of a functional.
The first example is a linear functional. The second example (because of the square) is a non-linear functional.

 

[zhermes]

$$f\mapsto f(a)$$
$$\int_a^b f(x)^2\, dx$$

Thank you for your reply!
But, I still don't understand what's special about these equations (for example); why aren't they just functions?
i.e. some function g(a,b) s.t.
$$g(a,b) \equiv \int_a^b f(x)^2\, dx$$
You're plugging in some 'a' and 'b', and getting a result 'g'... no?

 

[arkajad]

No, that is not the point. The correct notation should be:

$$F_{a,b}(f)=\int_a^b f(x)^2\,dx$$

Here $$F_{a,b}$$ is the functional. Its value on the function f is the calculated number. You change the function f - the number changes. The function f is the variable here. It varies in the space of all functions producing a different (in general) number for each function.
You should imagine the space of all functions - it is infinite dimensional, and draw a surface over this space.

P.S. I was not perfectly mathematically precise. I am trying to give you an idea.

 

[zhermes]

The function f is the variable here.

So the key is that the 'functional' is acting on a function, instead of a vector or scalar?
If so, how does a functional map from (e.g.) a vector space to its underlying scalar field?

 

[Tac-Tics]

The nomenclature is fluid, but the key idea is that the function's argument is not a real/complex number.

 

[arkajad]

If so, how does a functional map from (e.g.) a vector space to its underlying scalar field?

Indeed, as noticed above, the nomenclature is fluid. The main thing here is that you can add two functions f and g to make a new function, and you can multiply a function f by a constant number (scalar) c to get another function. Thus, with these operations, functions (on a given domain) form a vector space. Think of a function as a "vector" in an infinite dimensional vector space. Then a functional assigns a number to each vector in this space.

But, in general, apart of variational calculus, any linear map from vectors to numbers is called a "functional", a "linear functional". Assign to each vector its length - you have an example of a non-linear functional.

 

[zhermes]

Hmm, I see. Thanks! this has been very insightful.

 

[Landau]

Mathematically speaking, a functional is a special kind of function. A function in the general sense is just something that assigns to every element of some set A a unique element of some set B. But this is not what your text means. For them, I guess a function must have domain and codomain R or C.