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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?

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    $\begingroup$ None. ${}{}{}{}$ $\endgroup$ – copper.hat Mar 30 '16 at 16:53
  • $\begingroup$ I'd suppose a linear function takes values in $\mathbf R$ – in which case it would be better known as linear form. $\endgroup$ – Bernard Mar 30 '16 at 16:58
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    $\begingroup$ When I was in school, a linear function was just a line, i.e., $f(x) = ax+b$. In general, a function usually maps to $\mathbb R$ or $\mathbb C$, whereas a map or mapping can map to anything. $\endgroup$ – Friedrich Philipp Mar 30 '16 at 17:01
  • $\begingroup$ For some people, none. But for example $ax+by+c$ is traditionally called a linear function, though if $c\ne 0$ it should more precisely be called affine. $\endgroup$ – André Nicolas Mar 30 '16 at 17:12
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    $\begingroup$ Relevant, almost dup: math.stackexchange.com/questions/95741/… $\endgroup$ – leonbloy Mar 30 '16 at 17:57
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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.

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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.

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  • $\begingroup$ How are those two last not functions? $\endgroup$ – Tobias Kildetoft Mar 30 '16 at 18:05
  • $\begingroup$ Roughly speaking functions map points to points, Functionals map vectors to points and operators maps vectors to vectors $\endgroup$ – çiçek Mar 30 '16 at 18:07
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    $\begingroup$ None of those are meaningful distinctions. $\endgroup$ – Tobias Kildetoft Mar 30 '16 at 18:10
  • $\begingroup$ If you go with the somewhat naive definition that a function is something that takes an input and gives an output, functionals and operators do exactly that. If you take the better definition that a function is a certain special kind of relation, then functionals and operators are exactly that. I'm afraid that this answer is nonsense. $\endgroup$ – user296602 Mar 30 '16 at 18:33
  • $\begingroup$ Not just an input but a point, that has no directions and no movements however the others do that is what I meant when I used the term vector A function maps a real to a real (or complex), functional maps a set from function space or sequence space to a real (or complex). However, the operators is from a sequence or fonction space to sequence or fonction space $\endgroup$ – çiçek Mar 30 '16 at 18:36

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