A linear transformation is any transformation $f:U\to V$ between vector spaces over $\mathbb F$ for which
- $f(x+y)=f(x)+f(y)$
- $f(\alpha x) = \alpha f(x)$
for all $x,y\in U$ and all $\alpha\in\mathbb F$.
An affine transformation is any transformation $f:U\to V$ for which, if $\sum_i\lambda_i = 1$, $$f(\sum_i \lambda_i x_i) = \sum_i \lambda_i f(x_i)$$
for all sets of vectors $x_i\in U$.
In effect, what these two definitions mean is:
- All linear transformations are affine transformations.
- Not all affine transformations are linear transformations.
- It can be shown that any affine transformation $A:U\to V$ can be written as $A(x) = L(x) + v_0$, where $v_0$ is some vector from $V$ and $L:U\to V$ is a linear transformation.
Take an example where $U=V=\mathbb R^2$. Then $$f:(x,y) \mapsto(-2x+y, 3x+8y)$$ is a linear transformation, since
$$f((x_1,y_1)+(x_2, y_2)) = (-2(x_1+x_2) + y_1+y_2, 3(x_1+x_2) + 8(y_1+y_2)) = \\
= (-2x_1 + y_1, 3x_1 + 8y_1) + (-2x_2 + y_2, 3x_2 + 8y_2) = f((x_1,y_1)+f((x_2, y_2))$$
However, $$g:(x,y)\mapsto (-2x+y+5, 3x+8y-2)$$
is not a linear function (you can immediatelly see this since $g((0,0)) \neq (0,0)$, while linear functions always map $0$ to $0$).
Both $g$ and $f$ are (you can check) affine functions.
I honestly think trying to encode the topology into the terminology "linear space" is weird. I've always seen "vector space" and "linear space" used synonymously, and I think these are appropriately named sense since "vector space" suggests we're talking about a set of vectors (and vectors are to be added and scalar-multiplied), and "linear space" suggests a set with some structure of "linearity". So, in every situation I've seen, they just refer to the purely algebraic structure $(V,+,\cdot, \Bbb{F})$.
If we want to talk about a structure on top of the algebraic structure alone, then our terminology should reflect that explicitly. For example, if we wish to talk about a topology on top of the algebraic structure, we should speak of "topological vector space" or "topological linear space" (I've actually never seen this second term before, but I am sure people will know exactly what you mean). Furthermore we may then wish to talk about further structure, such as having a norm, in which case we'd speak of "normed vector space" or "normed linear space". If we wish to specify further properties, such as completeness, then we'd speak of "complete topological vector space" or "complete topological linear space" also one talks about "complete normed vector space" or "complete normed linear space" (i.e Banach spaces). We can also add about other adjectives, such as "complete, locally-convex, metrizable, topological vector/linear space" (i.e Frechet space).
Trying to unnecessarily suppress adjectives, and subsume them into a definition, is just weird to me (especially when it is already extremely common to use the terms "vector space" and "linear space" synonymously). Anyway, if your book defines things in one way, fine, but just know that it doesn't hurt to be more explicit in the terminology especially when you're communicating with others (after all, a large part of math is about communicating ideas).
Best Answer
A linear function (or functional) gives you a scalar value from some field $\mathbb{F}$. On the other hand a linear map (or transformation or operator) gives you another vector. So a linear functional is a special case of a linear map which gives you a vector with only one entry.