The question is about a proposition that leads to definition of affine morphism, in section 2.3 of "Geometry I" by Marcel Berger et al. To be searchable, I type the proposition and related text as follows:
Let $(X,\vec X, \Theta)$ and $(X',\vec {X'}, \Theta')$ be two affine spaces (over the same field), and $f:X\to X'$ a map (in set-theoretic sense). The following conditions are equivalent:
i) $f\in L(X_a; X'_{f(a)})$ for some $a\in X$;
ii) $f\in L(X_a; X'_{f(a)})$ for all $a\in X$;
There are more to come in the proposition, but I wish to prove i)$\Rightarrow$ii) first, with the hope that this proof can help me deal with the remaining. If the notations are not familiar to you, in affine space $(X,\vec X, \Theta)$, $X$ is the affine space, $\vec X$ is the underlying vector space, $\Theta$ is the map from $X\times X$ to $\vec X$ decided by simple transitivity. $L(E;F)$ is the set of all linear transformations from vector space $E$ to $F$. $X_a$ is the vectorialization of $X$ at $a$ (cf 2.1.9 of this text).
The first question is that I don't know if my understanding of vectorialization is correct. When writing $X_a$, does it mean the vector space structure is given, though I don't know what the concrete vector addition and scale multiplication is? If so, does the condition i) mean that, for some $a\in X$, $X$ is endowed with a vector space structure which is isomorphic with $\vec X$ under the map $\Theta_a$, while at the same time, $X'$ is endowed with a vector space structure which is isomorphic with $\vec{X'}$ under the map $\Theta'_{f(a)}$?
The second question is about the proof. To establish ii) under arbitrary $b\in X$, I think I have to construct a vector space structure on $X$ so that $X$ is vectorializable at $b$, as well as the same thing for $X'$ at $f(b)$. But how to do it? I just have no idea what first step to take from the given vectorialization. After that, I need to prove $f$ is a linear transformation between $X_b$ and $X'_{f(b)}$. But before I can endow the two affine spaces with a vector space structure, I don't know how to proceed.
I hope I have formulated my question clearly. Please let me know if there is any clarification needed. Thank you in advance for you help with the proof.
PS, perhaps a little off-topic, is there a text that have the same coverage, depth and rigor as that of Marcel Berger's book, but more accessible to nonmathematicians? Thanks for your recommendation.
I solved the problem myself, so I post the answer here. For the first question, let me assume my guess is correct. That is, when we write $X_a$, the vector space structure is given and therefore we can assume the vector space (and its operations) can be denoted $(X_a, +_a, \cdot_a)$. This vector space is isomorphic with $\vec X$ under $\Theta_a$. So, we are given two vector spaces $(X_a, +_a, \cdot_a)$ and $\bigl(X'_a, +'_{f(a)}, \cdot'_{f(a)}\bigr)$ in i).
At first, I had thought the vector space structure remains the same when $b\in X$ changes. This incorrect assumption led to an additional vector $\vec{ab}$ which made the additive linear relation in the isomorphism unbalanced. As a result, I later decided to fix this unbalance by adding a term in $X$ to the addition to make up an equality. Specifically, I define the vector addition corresponding to $b$ to be $$x+_by=x+_ay+_a\Theta_a^{-1}(-\vec{ab}).$$ Then I tried to define scale multiplication for $X_b$ by changing the scale, but this attempt seemed not working because there was always an additive term that I could not remove. So I went back to fix it by adding a term, and it turns out that the scale multiplication in $X_b$ should be $$c\cdot_bx=c\cdot_ax+_a\Theta_a^{-1}(-c\vec{ab}).$$ Note that the above two operation formulas are for vector space structures in $X$. For vector space structures in $X'$, we need to change them accordingly, say, by adding primes.
With the relation $\Theta_b(x)=\Theta_a(x)-\vec{ab}$ (the Chasles'relation), it can be verified that for $\forall b\in X$ ($a$ is fixed), the above formulas indeed define a vector space structure $X_b$ and make it isomorphic to $\vec{X}$ under $\Theta_b$. Similarly, $(X'_{f(b)},+'_{f(b)},\cdot'_{f(b)})$ is the desired vector space structure for $X'$ at $f(b)$ with the isomorphism being $\Theta'_{f(b)}$.
Next, with the given condition i) which can be translated into $f(x+_ay)=f(x)+'_{f(a)}f(y)$ and $f(c\cdot_ax)=c\cdot'_{f(a)}f(x)$, we can verify that $f$ is indeed a linear transformation between two vector spaces $(X_b, +_b, \cdot_b)$ and $(X'_{f(b)},+'_{f(b)},\cdot'_{f(b)})$ defined in previous paragraph. The only possible obstacle in the verification might be the equality $f(\Theta_a^{-1}\bigl(-c\vec{ab})\bigr)={\Theta'}_{f(a)}^{-1}\bigl(-c\overrightarrow{f(a)f(b)}\bigr)$. To show this, notice that since $\Theta_a$ is an isomorphism, $\Theta_a^{-1}$ is an isomorphism too. So, we have $\Theta_a^{-1}(-c\vec{ab})=-c\cdot_a\Theta_a^{-1}(\vec{ab})=-c\cdot_ab$, and hence $f\bigl(\Theta_a^{-1}(-c\vec{ab})\bigr)=f(-c\cdot_ab)=-c\cdot'_{f(a)}f(b)$. For the same reason, ${\Theta'}_{f(a)}^{-1}$ is an isomorphism so ${\Theta'}_{f(a)}^{-1}\bigl(-c\overrightarrow{f(a)f(b)}\bigr)=-c\cdot'_{f(a)}{\Theta'}_{f(a)}^{-1}\bigl(\overrightarrow{f(a)f(b)}\bigr)$ which is just $-c\cdot'_{f(a)}f(b)$.
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I have just understood vectorialization defined in 2.1.9 of the text. The vector space structure assigned to $X_a$ is not something given but unknown, but can be constructed (uniquely) based on the affine space $(X, \vec X, \Theta)$. Specifically, for any $a\in X$, define $x+y=\Theta_a^{-1}(\vec{ax}+\vec{ay})$ and $cx=\Theta_a^{-1}(c\vec{ax})$. It can be verified that it is indeed a vector space and is isomorphic with $\vec X$ under $\Theta_a$. Note that in general the vector space structure changes as $a$ changes. Fortunately, it can proved that the operations defined in my original answer are consistent with the vector space structure just described, so it is still valid.