# The similarities and differences between *function(s)*, *mapping(s)*, *operator(s)*, projection(s)*?

I just begin to study math books in English but I am not a native speaker. I am confused about the following mathematical terms:

function(s), mapping(s), operator(s), projection(s)

Is there anyone can explain in details about the similarities and differences between these terms, and what mathematicians want to emphasize when they use them?

PS. I think they are very similar and sometimes interchangeable to each other, because they all imply some relations pairing from one set to another set. Briefly speaking, in my opinion, mapping just means a connection, function is a specific mapping from input set to output set but the output object is restricted to be unique, operator is a specific mapping from space to space (while space is a specific set?), projection sounds like mapping. In summary, function $\subset$ functional $\subset$ operator $\subset$ mapping $\equiv$ projection, OR functional $\subset$ operator $\subset$ function $\subset$ mapping $\equiv$ projection?

• The terms take different meanings in different contexts; can we assume that you're talking specifically about linear algebra and functional analysis? Commented Aug 23, 2017 at 15:06

A function assigns each element of the domain to a unique element of the codomain.

In my experience mapping and function are interchangeable.

An operator is usually a function from a vector space to itself. Often but not always it is linear. Very often the space in question is infinite dimensional.

A projection is usually an idempotent linear function. Idempotent means that $P(P(x))=P(x)$. (Thus a projection always maps a space to itself.) When an inner product is available, we often look at orthogonal projections, which for projections just means self-adjoint.

A functional, in analysis at least, is usually a function from a vector space to its base field. It gets its name because often the vector spaces in question have functions as their elements; for example, (definite) integration is a functional on a vector space of real-valued functions on some domain. These are often but not always linear.

A map or mapping is a relation $R$ such that $aRx$ has always one and only one solution where $a$ is in the domain of the relation.

A function is a map from a numerical set $X$ to a numerical set $Y$. $X$ and $Y$ can be subset of $\mathbb{R}$, $\mathbb{C}$ (or other set of numbers) or their cartesian powers. Then you have "scalar functions", "vector-valued functions", "multivariable functions", ... Function can also be used to mean map.

A functional is like a function where the restriction on the kind of set pertains to the codomain $Y$ only. Then you have "scalar functional", "vector-valued functional". In particular when $X$ is a vector space over some field $\mathbb{K}$ then it is assumed that $Y=\mathbb{K}$ for scalar functional, $Y=\mathbb{K}^n$ for vector-valued functional. A functional over $X^n$ for a given set $X$ is usually called an $n$-form over $X$.

An operator is usually an endofunction, that is a map from a set $X$ to itself. It can also be a $n$-ary operator (i.e. binary operator) over $X$, that is, a map $X^n\rightarrow X$. In this sense the former meaning can be distinguished from this latter saying unary operator.

A projection is the map that takes an element of a set $X$ into an equivalence class of this element w.r.t. some equivalence relation on $X$.