Problems with this reasoning (sum of subspaces) Let be $E$ a $\mathbb{K}$-Vector space and $U,W$ subspaces of $E$. Prove that $S=U+W$ It is the intersection of all subspaces containing U and W.
My idea:
If $U\cup W$ is subspace of $E$ then $U\subseteq W$ or $W\subseteq U$ if that happen we have $U \cup W=U+W$. Now if $x\in U+W$ then $x=u_1+w_1$, for some $u_1\in U$ and $w_1\in W$, but $x=1(u_1)+1(w_1)$ this is a linear combination of vectors in $U+W$, then $x\in span (U\cup W)$
I do not know if this reasoning is quite right, I have many doubts, if you could please help me, it would be great. Thank you!
 A: HINT: Let's name $K=\bigcap_{\lambda\in \Delta} G_{\lambda}$ the intersection of all subspaces containing $U$ and $W$. Let $S$ be the sum of U and W, this is $S=U+W$
Now you can prove that $K\subset S$ and $S\subset K$ This will get you that $K=S$.
Let $x\in S$ $\Rightarrow$ $x\in U+W$ $\Rightarrow$ $\exists u_1\in U,                                                              {w_1} \in W$ such that $x=u_1+w_1$ , because $ G_{\lambda}$ is a linear subspace that contain both $U$ and $W$  implies that $u_1+w_1\in G_{\lambda}$ $\forall$  $\lambda$ $\Rightarrow x\in G_{\lambda}$ $\Rightarrow x\in
 \bigcap_{\lambda\in \Delta} G_{\lambda}$ and that gets you this $S\subset K$
I'll leave the other part to you. Good luck!
A: Let $V$ be the intersection of all subspaces of $E$ containing $U$ and $W$. We want to prove that $U+W\subseteq V$ and $V\subseteq U+W$.
Let $Z$ be a subspace of $E$ containing $U$ and $W$. In particular, for every $u\in U$ and $w\in W$, we have $u+w\in Z$. Therefore $U+W\subseteq Z$. Thus $U+W$ is contained in $V$.
On the other hand, $U+W$ is a subspace containing $U$ and $W$, so $V\subseteq U+W$.
