Show that $A + B$ is a subspace of $H$ that contains $A$ and $B$ Please hand tips and hints in solving:
Let $H$ be a vector space.
For $U_1, U_2$ subsets of $H$, define $U_1 + U_2$ to be the set $\{u_1 + u_2\,|\,u_1\in U_1 \wedge u_2 \in U_2\}$. Let $A$ and $B$ be subspaces of $H$.
$a)$ Show that $A + B$ is a subspace of $H$ that contains $A$ and $B$.
 A: A subset $S \subset H$, where $H$ is a vector space, is a subspace of $H$ if and only if: (i) $0\in S$, (ii) $\alpha x + \beta y\in S$ for every $x,y\in S$ and $\alpha,\beta \in\mathbb F$, where $\mathbb F$ is the field that underlies the vector space $H$.

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*$A+B$ contains $A$ and $B$: Certainly $A \subset A + B$ and $B \subset A+B$ by definition of $A+B$, since $0\in A$ and $0\in B$ (they are subspaces).


*$A+B$ is a subspace: To show that $A+B$ is a subspace of $H$, note first that $0\in A+B$ since $0\in A$ and $0\in B$. Also, if $a_1 + b_1\in A+B$ and $a_2 + b_2\in A+B$ where $a_1,a_2\in A$ and $b_1,b_2\in B$, we have $\alpha (a_1 + b_1) + \beta(a_2+b_2)\in A+B$ for every $\alpha,\beta\in\mathbb F$ (the field over which $H$ is a vector space). To see this, observe that $\alpha (a_1 + b_1) + \beta(a_2+b_2) = (\alpha a_1 + \beta a_2) + (\alpha b_1 + \beta b_2)$. Since $A$ is a subspace, $\alpha a_1 + \beta a_2 \in A$. Since $B$ is a subspace, $\alpha b_1 + \beta b_2 \in B$.
I have left some details for you to fill, and I hope you're able to take it from here. If you have any questions, please let me know.
A: Let $H$ be a vector space over some field $\mathbb{K}$ and let $A$ and $B$ be subspaces of $H$. We want to prove that $A+B$ as you defined is a subspace of $H$. Two things need to be notice. For $A+B$ be a subspace of $H$,
$(i)$ the set $A+B$ must be a subset of $H$; (and it is in fact, try to understand why)
$(ii)$ $A+B$ needs to be closed for the operations defined for $H$ and restricted to $A+B$.
The second point comes straight for the definition, but we also have a characterisation which is very handful to prove this sort of things. I will leave it here (I guess that you might already know it, but if you not, you should attempt to prove).
$\to$ Let $V$ be a vector space and $\emptyset \neq U \subseteq V$ with the operations defined in $V$ but restricted to $U$. Then, $U$ (together with these operations) is a subspace of $V$ if and only if $\alpha u + \beta v$, for all $u,v \in U$ and $\alpha, \beta \in \mathbb{K}$.
These hints should be sufficient for your work.
