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Function vs. Functional?

  1. Sep 7, 2010 #1
    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!
     
  2. jcsd
  3. Sep 7, 2010 #2
    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

    [tex]f\mapsto f(a)[/tex]

    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

    [tex]\int_a^b f(x)^2\, dx[/tex]

    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.
     
  4. Sep 7, 2010 #3
    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.
    [tex]
    g(a,b) \equiv \int_a^b f(x)^2\, dx
    [/tex]
    You're plugging in some 'a' and 'b', and getting a result 'g'.... no?
     
  5. Sep 7, 2010 #4
    No, that is not the point. The correct notation should be:

    [tex]F_{a,b}(f)=\int_a^b f(x)^2\,dx[/tex]

    Here [tex]F_{a,b}[/tex] 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.
     
  6. Sep 7, 2010 #5
    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?
     
  7. Sep 7, 2010 #6
    The nomenclature is fluid, but the key idea is that the function's argument is not a real/complex number.
     
  8. Sep 7, 2010 #7
    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.
     
  9. Sep 7, 2010 #8
    Hmm, I see. Thanks! this has been very insightful.
     
  10. Sep 8, 2010 #9

    Landau

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    Science Advisor

    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.
     
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