Definition about sum of subspaces

Question

I'm a little confused about definition of sum of subspaces.

Suppose $U_1,U_2, \ldots , U_m$ are subspaces of $V$ and $S=U_1+U_2+ \ldots +U_m$ is not a direct sum. If there is a vector $v$ in $S$ that can be written in two ways $v=u_1+u_2+ \ldots + u_m=v_1+v_2+ \ldots v_m$ where $u_i,v_i \in U_i$. Then, if we try to express $S$, do we have write $v$ twice like $S=\{ \ldots, v,v, \ldots \}$ or just one time $S= \{ \ldots , v, \ldots \}$ is fine?

Furthermore, if we are trying to prove $S=T$ for some subspace $T$, do we need to prove that $v$ also appears twice in $V$ ? Or only prove $v \in T$ is fine?

Answer 1

The notation $\{, \}$ (i.e., curly braces) is a standard set notation, implying that any duplicates are only written once.

In this case, the term "duplicates" may be a bit blurred since there are multiple ways of writing the same vector twice in terms of decomposing it as a sum of other vectors, nevertheless, they are functionally the same, so $\vec{v}$ is written out only once despite having multiple methods of creating it.