This is a well known and repeated question but I would like to dig a bit deeper in it.
When proving this we need to verify each property.
Let us say we have already proved it is closed under + and scalar multiplication.
We want now to prove associativity:
Let $S, T, Q \in L(V, W)$. This means each of them is a linear map from the vector space V to the vector space W). So it makes sense to say $S(x)$ is an element of $W$ for every $x \in V$.
Associativity:
$ S + (T + Q) = (S + T) + Q$
The standard proof uses that $[S + (T + Q)] (v) = [(S + T)+Q] (v)$
So it means we consider the same $v \in V$.
My question is: why doing
$S(v_1) + (T(v_2) + Q(v_3)) = (S(v_1) + T(v_2))+Q(v_3)$
would be wrong? Where $v_1, v_2, v_3 \in V$.
The last equation would mean $ w_1 + (w_2 + w_3) = (w_1 + w_2) + w_3$ which is true because $W$ is a vector space.
I believe the mistake lies in the fact that we have defined $(S+T)v = Sv + Tv$ and proved that $S+T$ is linear map so when we are proving associativity:
Let $T+Q = A$ and $S+T = B$,
we are doing $(S + A)v$ and we want to see if this is the same as $(B + Q)v$
It would be nice to read some wisdom that clarifies the whole scenario.
Thanks as always!
Best Answer
Let's go back for a second, when we are proving associativity for example on a monoid $(M, +)$ we want $$\forall a,b,c \in M,\space \space a+b+c = (a+b)+c = a +(b+c)$$ In this situation we don't have to worry about $a,b,c$ being linear functional we just need to find a way to verify the associativity holds. In our example we want $\forall S,T,Q \in L(V,W)$ that associativity holds, what this means? That means, they have to be the same as linear functional, since they are elements in $L(V,W)$ $$(S+T) + Q = S+(T+Q)$$ how do we verify if two linear functionals $L_1, L_2$ over $V$ are the same, for example iff $L_1(v) = L_2(v), \forall v \in V$, in our case means $$[(S+T) + Q](v) = [S+(T+Q)](v)$$ each of them applied to the same $v$. Using three different vectors doesn't lead us to a wrong conclusion but we aren't verifying the associativity of $L(V,W)$ we are just verifying if that particular equation is true, which is for the associativity of $W$. Hope this helps!