# What's a functional?

1. Jan 9, 2007

### moonman

So far all I know is that a functional is a function that has a set of functions as its domain. So what does that mean?

I have a functional that looks like dy/dx = a bunch of constants.
What I'd like to know is how to take that and plot it. Can this be done?

2. Jan 9, 2007

### Tom1992

do you mean a linear functional? if you mean a linear functional, it is a mapping f from a vector space to its field such that

f(au+bv) = af(u) + bf(v).

the set of all linear functionals form a vector space called the dual space of V and has the same dimension of V. it's basis is called the dual basis.

if you mean some other kind of functional, then it is a function of a function. for example, in the "calculus of variations" the functional represents a function of a function and we want to find what form the second function must be so that the first function is a minimum or maximum. this function must satisfy the euler-lagrange equation

-d/dx(dy/df') + dy/df = 0,

which is a differential equation, so that the answer is the function we seek. (notice that y is a function of f, which itself is function of x, so y is a function of a function--a functional)

for example, the brachistochrone, problem: find the curve that minimizes the time spent for a particle to slide down the curve, the functional is the time which you want to minimize. it is a function of the curve (which is a function), you seek to find the curve so that the functional (the time) is minimum.

my daddy told me about the brachistochrone problem and the tautochrone problem last year. this is what i remember from him.

of the three types of analysis: real analysis, complex analysis, and functional analysis, i've been told that functional analysis is the most difficult. i'll dig into that textbook from my dad's library soon...

Last edited: Jan 9, 2007
3. Jan 9, 2007

### Hurkyl

Staff Emeritus
It means that it has a set of functions as its domain. *shrug* If F is a functional, and f is an element of its domain, then F(f) is in the range of F.

I really can't figure out what you're describing: could you present it in more detail?

4. Jan 9, 2007

### moonman

Sorry, there was a mistake with my original entry. The function is not a derivative. I'm working on a physics experiment, where the data has a divergence. In my research I found that there is a functional of the form y=constant that is commonly used to fit this sort of divergence. Supposedly it should look logrithmic.

My problem is that I have a functional y=constants, which is supposed to be plotted against my data. I just don't know what that means. How do you go from that equation to a logarith? If this isn't enough detail for you to help me, thanks anyways.

5. Jan 9, 2007

### HallsofIvy

Staff Emeritus
More generally, a "functional" is a function having a vector space as domain and the underlying field as range.

Of course, functions (with various conditions: continuous, differentiable, infinitely differentiable, integrable, etc.) form a vector space over the real (or complex) numbers and so a function that assigns a number to every such function is a functional. Examples would be $\frac{df}{dx}(a)$,the derivative of f at x= a (over the set of differentiable functions) or a definite integral such as $\int_a^bf(x)dx$ over the space of integrable functions.

However, in neither of those definitions can I make sense of "a functional y= constants" since I don't know what you mean by y.