TITLE: What is the precise difference between functions and operators?
QUESTION [7 upvotes]: 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?

REPLY [2 votes]: 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. $$f:\{1,2,3,4\}\to \{1,2\}$$