Difference between functional and function?

[aditya23456]

I know function is just a subset of functional but physical example helps to understand this difference.."Any physical situation" thanks in advance

 

[birulami]

As far as I understand it, a functional is specifically a function that maps other functions to real (complex, field) numbers.

 

[chiro]

It's pretty much what birulami said however you are dealing with things inside a vector space which has conditions like linearity and the behaviour of a cartesian geometry.

It might help you to look at inner products for l^2 and L^2 spaces which deal with normal inner products as a sum and inner products as an integral calculation. These operations will produce complex numbers in the general format.

It's the same kind of way of how we do inner or dot products with vectors to get a scalar value. This might give you a better understanding of the concepts:

http://en.wikipedia.org/wiki/Functional_(mathematics)

Also be aware that we have a framework in modern mathematics of a) defining operators that fit the linearity condition (known as linear operators) and a calculus has (and still is) being developed to deal with analyzing these under the perspective of a vector space, even if we have infinite-dimensions and b) a framework known as integral transforms which deal with a kind of 'generalized projection framework' for the L^2 spaces where we deal with Kernels in a general way instead of dealing with specific bases.

As an example, one integral transform is the Fourier transform. Another is the Haar-Wavelet. In general though, it can be anything that has the right conditions for both the function and the actual basis that we are 'projecting' too.

 

[SteveL27]

I know function is just a subset of functional but physical example helps to understand this difference.."Any physical situation" thanks in advance

I think functional is a subset of function. That is, a functional is a function with certain properties, typically a function that maps functions to numbers, in particular a linear function that maps functions to numbers.

 

[rjb]

A function is a mapping from one [1st] set to another [2nd] which may be equivalent (equal) to the 1st (eg y=x^2, x,y real number - a mapping "within" the set of real numbers. The elements of the sets may be numbers (scalars), vectors, other functions (a function of a function), matrices and so on
As I recall (at 82 I'm too lazy to look it up!), a functional is a mapping from a set to the real/complex numbers.