Suppose that \(U\) is a subspace of \(V\). What is \(U + U\)?
\(U + U = U\).
Take \(u, v \in U\). Then every element in \(U + U\) can be written as \(u + v\). Since \(U\) is a subspace of \(V\), U is closed under addition. Therefor \(u + v \in U\). Hence \(U + U \subset U\).
If we take \(w = u + 0 \in U + U\), we can get every element in \(U\) (\(w \in U\)). Therefor \(U \subset U + U\).
Thus \(U = U + U\).
I strongly recommend you to read the previous blog which discussed a similar problem. You can go to that blog by clicking link: 证明V的两个子空间的并是V的子空间当且仅当其中一个子空间包含另一个子空间. I will skip the same part of this proof which is discussed in that proof.
Prove that the union of three subspaces of \(V\) is a subspace of \(V\) if and only if one of the subspaces of \(V\) contains the others.
Let \(A\), \(B\), \(C\) be the three subspaces of V.
Assume \(A \cup B \cup C = A\) or \(B\) or \(C\).
Clearly \(A \cup B \cup C\) is one of these three vector spaces of \(A\), \(B\), \(C\). Therefor \(A \cup B \cup C\) is the subspaces of \(V\).
Assume \(A \cup B \cup C \neq A\) or \(B\) or \(C\).
Firstly, consider the condition:
\(A \cup B \cup C = A \cup B\) or \(B \cup C\) or \(A \cup C\).
We have discussed this situation in the previous blog, you can read it by clicking this link: 证明V的两个子空间的并是V的子空间当且仅当其中一个子空间包含另一个子空间.
Secondly, consider the situation that neither of these three subspaces contained in the others.