# Are operator and mapping the same concept?

1. I was wondering what differences and relations are between a mapping and an operator generally?

For topological vector spaces or functional analysis, it seems like an operator and a mapping are the same concept, doesn't it?

2. What differences are between an operator and an operation? Is an operation a mapping from $X^n$ to $X$ for some set $X$ and some $n \in \mathbb{N}$?

Thanks and regards!

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Mapping is generally another name for function, and an operator is often used to mean a binary function (one with two arguments) but occasionally (especially in functional analysis) used in place of the word function. – Alex Becker Feb 23 '12 at 23:58
@AlexBecker: Thanks! (1) what about an operation? (2) Do you mean that an operator and a binary operation are the same concept, generally except some special cases such as in functional analysis? – Tim Feb 24 '12 at 0:00
I've seen operator/operation used interchangeably, but I don't know if its proper. – Alex Becker Feb 24 '12 at 1:41

Historically, "function" meant something like an element of the vector space $C(\mathbb{R})$ of continuous functions $\mathbb{R} \to \mathbb{R}$, "functional" meant something like a linear functional $C(\mathbb{R}) \to \mathbb{R}$ (that is, a thing which takes functions as input and returns numbers), and "operator" meant something like a linear transformation $C(\mathbb{R}) \to C(\mathbb{R})$ (that is, a thing which takes functions as input and returns functions).