Reading linear algebra done right 3rd edition.
For $R$, concerning about empty set as additive identity probably does not make sense.
Though for \texttt{sum of subsets}, its definition is $$ U_1 + \dots + U_m = \{u_1 + \dots + u_m : u_i \in U_i \} $$
With this definition, it seems $U_1 + \emptyset$ is a valid addition operation, which is an addition simply without elements from other $u_i$.
I know this understanding is wrong. After all, if the above claim is true then $\emptyset$ would be an additive identity to any subspace. That is in contradiction with the fact that each subspace should only have one unique additive identity.
What do I miss?
Update:
As a reference, checkout the definition of "sum of subsets". it explicitly says "subsets", rather than "non-empty subsets", or "subspaces". Does that imply this definition allows empty set?
Best Answer
By the definition, $$ U_1 + \emptyset = \{u_1 + u_2 : u_1 \in U_1, u_2 \in \emptyset\}. $$ But there are no sums $u_1 + u_2$ where $u_1 \in U_1$ and $u_2 \in \emptyset$, because it's impossible to choose $u_2 \in \emptyset$. Therefore, $U_1 + \emptyset = \emptyset$. This is not what an additive identity does!
(Also, the sum $U_1 + \emptyset$ is not interesting in the context of linear algebra, because $\emptyset$ is not a vector space. But the addition of sets is defined in other settings too, so it's not wrong to ask what $U_1 + \emptyset$ is: you just don't get an interesting answer.)