# Difference between functional and function?

In summary: It's a subset of function because a function is a mapping from a set to itself, while a functional is a mapping from a subset of a set to the real/complex numbers.
I know function is just a subset of functional but physical example helps to understand this difference.."Any physical situation" thanks in advance

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

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.

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.

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.

## What is the difference between functional and function?

Functional and function are two terms that are often used interchangeably in scientific research. However, there is a subtle difference between the two.

## How do you define functional?

Functional refers to the purpose or role of something in a larger system or process. It is the intended function or use of something.

## What is the definition of function in science?

In science, function refers to the specific action or process that something performs. It is the way in which something operates or behaves.

## Can you provide an example of functional and function?

An example of functional could be the heart, which has the function of pumping blood throughout the body. Another example could be a pencil, which has the function of writing or drawing.

## Why is it important to understand the difference between functional and function?

Understanding the difference between functional and function is important in scientific research as it allows for more precise and accurate communication. It also helps in identifying the purpose and role of different components in a system, which is crucial in understanding how the system works.

Replies
2
Views
745
Replies
4
Views
824
Replies
0
Views
604
Replies
4
Views
1K
Replies
2
Views
337
Replies
9
Views
2K
Replies
5
Views
2K
Replies
1
Views
1K
Replies
2
Views
556
Replies
11
Views
742