If a linear space can be represented as the direct sum of two sets of subspaces, how do I show that the subspaces are equivalent?

Question

I need some help with proving the following.

"Let linear space $X = \oplus_{j}U_{j}$ where ${U_{j}}$ is a set of non-trivial subspaces of $X$. Let subspace $V_{j}$ be such that $X = \oplus_{j}V_{j}$ and that $U_{j} \subset V_{j}$. Show that $U_{j} = V_{j}$."

I'm pretty sure I understand the individual concepts, but I'm really struggling to put them together into a proof.

Answer 1

Fix $j$. Let $y \in V_j$. Then we can write $y =\sum x_i$ with $x_i \in U_i$ for all $i$. Note that $x_i \in V_i$ for all $i$. Another decomposition of $y$ as sum of elements of $V_i$'s is $0+0+...+y+0+...$ where $y$ is in the $j-$th place. By uniqueness we get $y=x_j$ Hence $y \in U_j$.