Is there any difference between additivity and linearity?

Question

Can an additive model be nonlinear? Can a linear model be nonadditive? Kindly provide examples and references.

Answer 1

If X and Y are vector spaces, we say a mapping T form X to Y is additive if T(x1 + x2) = T(x1) + T(x2) for every x1 and x2 in X.

We say T is linear if, in addition to the previous functional equation, we have T(a*x) = a*T(x) for every x in X and every scalar a. Therefore, every linear model is additive.

But not every additive model is linear, though the examples are somewhat patological.

If X and Y are real vector spaces, T is additive and you add the assumption that it is continuous, then T is linear.