Show that $H+K$ is a subspace of $V$ I am trying to solve this, please give me hints.
$H,K \in V$, $V$ is a vector space.
$H+K=\{w:w=u+v:u \in H, v \in K \}$
Show that $H+K$ is a subspace of $V$.

A sufficient condition is that $H$ and $K$ are subspaces of $V$. It suffices to check if we have $aw_{1} + bw_{2} \in H+K$ for all scalars $a,b$ and all $w_{1},w_{2} \in H+K$.
Let $w_{1},w_{2} \in H+K$ and let $a,b$ be scalars. Then $w_{1} = h_{1} + k_{1}$, $w_{2} = h_{2}+k_{2}$ for some $h_{1},h_{2} \in H$ and some $k_{1}, k_{2} \in K$ by definition. Thus $aw_{1} + bw_{2} = ah_{1} + bh_{2} + ak_{1} + bk_{2}$. But $ah_{1}+bh_{2} \in H$ and $ak_{1} + bk_{2} \in K$  by assumption, so by definition we have $aw_{1} + aw_{2} \in H+K$.
Note that if $H,K$ are simply subsets of $V$, then $H+K$ need not be a subspace of $V$; for if $V := \mathbb{R}(\mathbb{R})$, $H:= \{ 0 \}$, and $K := \{1\}$, then $H+K = \{ 1 \}$, which is not a subspace of $V$. 
