My attempt : for a)

First I notice that $\displaystyle W \subset V$. Now I must show that the addition of 2 vectors in $\displaystyle W$ remains in $\displaystyle W$ and the multiplication of a vector by a scalar remains in $\displaystyle W$. I understand it's obvious because multiplying a vector $\displaystyle w$ which is orthogonal to $\displaystyle v$ will give another vector orthogonal to $\displaystyle v$.

Let $\displaystyle p$ be the inner product function. We have the property : $\displaystyle p(w,v)=0$. And I see that $\displaystyle p(kw,v)=kp(w,v)=k\cdot 0=0$.

Now I must show that $\displaystyle (w_1+w_2)\in W$ $\displaystyle \forall w_i \in W$.

$\displaystyle p(w_i+w_j, v)=p(w_i, v)+p(w_j, v)=0+0=0$.

Wow, it seems I've show part a)! Eventually I must show other properties of $\displaystyle W$, like that the zero vector is in... and so on.