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I have heard affirmatively that all operators are functions, but not all functions are operators.

But at the same time I have heard that functions map numbers to numbers, whereas operators map functions to functions.

But if operators are function, then a function map functions to functions. How do you untangle this mess?

Can someone present a definitive difference between functions and operators?

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    $\begingroup$ There are informal differences of usage, but no formal difference. $\endgroup$ – André Nicolas Jul 25 '15 at 22:05
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    $\begingroup$ I would broaden the question to What is the difference between a map, a function, an operator and a morphism :) $\endgroup$ – A.Γ. Jul 25 '15 at 22:08
  • $\begingroup$ I think your question is similar to "what is the difference between a 'family,' 'collection,' and 'set?'" The answer is that we don't always want to talk about sets whose members are sets of sets containing sets... And so on. Using several terms for the same thing may in some cases obfuscate things but in many cases it also helps to clarify a tangled web of definitions. $\endgroup$ – walkar Jul 25 '15 at 22:13
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    $\begingroup$ At an elementary level, functions generally map numbers into numbers, whereas operator work on more abstract things such as elements of a vector space, other functions, symbols, etc - this is not a formal definition, the terms have a lot of overlap, so it would not be incorrect to say a function works on matrices for instance. It seems a bit strained to talk about operators working on numbers, but operator is used in that context in various branches. You'll find each subject has its own favorite terms which have equivalents in other areas. $\endgroup$ – user247608 Jul 25 '15 at 22:26
  • $\begingroup$ Duplicates? math.stackexchange.com/questions/168378/… math.stackexchange.com/questions/331382/… $\endgroup$ – A.Γ. Jul 25 '15 at 23:43
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According to wikipedia, an operator is a function whose domain and codomain are both vector spaces or modules.

Since $\mathbb{R}, \mathbb{Q}, \mathbb{C}$ are all (one-dimensional) vector spaces, many familiar functions are also operators. However, a general function might be from a domain that is not a vector space, and hence not be an operator, e.g. $$f:\{1,2,3,4\}\to \{1,2\}$$

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  • $\begingroup$ I can think of a number of operators which have nothing to do with vector spaces, that wiki article is not accurate - even wiki refers to the boundary operator of simplices. $\endgroup$ – user247608 Jul 25 '15 at 22:30
  • $\begingroup$ I'm no topologist, but it seems that simplices can be thought of as elements of a module. $\endgroup$ – vadim123 Jul 25 '15 at 22:47
  • $\begingroup$ You (and the wikipedia article you cite) seem to be a bit confused about the difference between sets and structures. It is a category error to say that $\{1, 2, 3, 4\}$ is not a vector space: a vector space is a structure satisfying certain axioms but $\{1, 2, 3, 4\}$ is just a set (which can be the underlying set of a vector space over the field with two elements). $\endgroup$ – Rob Arthan Jul 25 '15 at 23:36
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    $\begingroup$ The set $\{1,2,3,4\}$ is not a vector space, absent imposing additional structure. Hence if we do not choose to add that somewhat unnatural structure, $f$ is a function but not an operator. $\endgroup$ – vadim123 Jul 26 '15 at 2:17
  • $\begingroup$ You've missed the point about the category errors that both you and the wikipedia article are making: you are conflating the underlying set of a structure with the structure itself. It is meaningless to say that a function whose domain is $\{1, 2, 3, 4\}$ is not an operator because its domain is not a vector space: $\{1, 2, 3, 4\}$ does not have the right form to be a vector space, but it can certainly be the underlying set of a vector space. See en.wikipedia.org/wiki/…. $\endgroup$ – Rob Arthan Jul 26 '15 at 20:19

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