What is the difference between linear mappings and linear functions? Let $V$ and $V'$ be vector spaces over a field $K$. A linear mapping $$f:V \to V'$$ is a mapping which preserves addition and scalar multiplication.
My question is: what is the difference between linear mappings and linear functions? 
 A: Depends on the definition you are using. Linear mappings are typically the same as linear functions, although in some contexts, a linear function is strictly some function in the form of $y = mx+b$. 
Linear mappings are functions that map two domains and whose operations have linearity, that is the linearity of scalar multiplication and addition.
A: For the definition of linearity it still the same for functions or mappings.
But the term mapping has more general uses than than function, a mapping could be a function, functional or operator... 
However, these three terms are not the same.
 A function is a term you used for mappings that map a set A to B where A and B are subsets of real of complex numbers or almost any map into $R^n$ or $C^n$ .
A functional is a mapping that is defined from a subset of a space X into the space of real or complex numbers.
 Finally an operator is defined from a space X to a space Y, where X, Y could sequence or function spaces.
Here an examples that clearfy more the difference between these three terms, 
let the function $f:R \rightarrow R $ such that  $f(x)= x$. 
Let us integral this function between $0$ and $1$,  then $ \int^{1}_{0}x =\frac{1}{2}$.
You can notice that the defined integral is a functional, it mapped a function ( x ) into a real ( 1/2).
However the undefined integral of this function is a function,  $\int x =\frac{1}{2}x^2$.
Thus, the undefined integral is an operator.
You can use the term of mapping for all previous examples but the term function can be uses only for the first case.
