Difference in usage between function, mapping, functional, form, and operator?

Question

The word function has many synonyms (or close to synonyms), including:

• map
• functional
• form
• operator
• transformation

What is the difference, in meaning or usage, between them? I understand that exact definition may not be standard. But there certainly seems to be a difference in typical usage. For example, multilinear form: what is form in that context, and how does it typically differ from function?

What is the difference in typical usage of each quasi-synonym for function?

Answer 1

At the high-level , all are Exactly Same , in the general sense that , a function "converts" the given "Input" into Exactly 1 "Output".

In Particular Cases , there are variations on what the "Input" is , what the "Conversion" is trying to achieve , what the "Output" is , hence there are many Synonyms.

Map : In terms of relations , we can "connect" elements of Set X (Domain) with elements of Set Y (range) , which gives us tuples like $xRy$. The Collection of these tuples is given the name Map or Mapping.

Calculus of Variations : A functional is a type of function which takes a function to generate a real number , using Integration between 2 Implicit limits.

$F(f) = \int^{A}_{B} f dx $

$F(x^2) = \int^{A}_{B} x^2 dx $

Operator theory : We may treat Differentiation in very general terms like $Dx=1$ & $D^{2} x^{n}=n(n-1)x^{n-2}$ where we have "invented" a new Operator D which takes a function to generate a function.
In more general terms , a Binary Operator $O$ operating on 2 elements $x$ , $y$ to generate $z=xOy$ , may be treated like a function $O(x,y)=z$

Cartesian Geometry / Vector Calculus / Complex Numbers / Matrix Calculus : given a Point $X$ in terms of Vector or Complex Number , we can generate or transform that Point into a new Point $Y$ by some Conversion Process , involving other vectors Complex Numbers or Matrixes , eg $AX=Y$ or $Y=cX$. We still have a function , which we name a transformation.
We may want to think about converting Data (given in terms of a Discrete functions or real number or vectors) into Data in some other format (in terms of Continuous functions or trigonometric functions or Complex numbers) , where we may utilize some suitable transformations , like z transform or laplace transform or fourier tranform.

Norm : A norm converts a vector or Matrix to a real number. The Norm "function" may be a way to think about the magnitude of the vector or Matrix.

A vector field is a Mapping from $R^n$ to $R^n$

Likewise , we have linear forms (which takes a vector or matrix X to generate AX+B) , Bilinear forms (which takes 2 vectors or matrixes to generate a linear Combination) in Linear Algebra.
Here , a form is a Category or Class. It just means a "format" of the terms involved. Distinguishing between $AX$ & $AXX$ , it is a linear format or a Quadratic format.

There are a lot more Synonyms , all of which have the Commonality with functions.
There are many Differences in each Case , which will warrant a new name , yet there are enough Similarities to show that these are still variations of functions.

Each Synonym is used in Specific Areas with Specific Semantics. It will not make much sense outside of that Area.
A transformation will not convert a vector to a matrix , a functional will not convert a real number into a function , a vector field will not convert a function to a Set , a norm will not convert a rational number to a Matrix.

Answer 2

The following is true in most (modern) contexts:

Function and map/mapping are usually synonyms, both words are used to refer to any function (in the set-theoretic sense) $f:A\to B$ between sets $A$ and $B$. However, some authors in some contexts reserve the word function for the case in which $B=\mathbb{R}$ or $\mathbb{C}$, and use the term mapping for the general concept.

Functional and form are often synonyms, they both refer to a mapping from a $K$-vector space $V$ to its scalar field $K$. In your case, a multilinear form is just a multilinear mapping from a product of vector spaces to their scalar field.

Operator is generally used to refer to a mapping between normed vector spaces.

Finally, transformation is perhaps the least agreed upon. From my experience, most authors use it for bijective mappings that preserve some kind of structure (usually linear or affine, as in linear or affine transformations).

I have to remark that this is based on my own personal experience. Many might disagree. Some authors use the terms slightly differently, so, in the end, one must always check the terminology.