My attempt : for a)

First I notice that

. Now I must show that the addition of 2 vectors in

remains in

and the multiplication of a vector by a scalar remains in

. I understand it's obvious because multiplying a vector

which is orthogonal to

will give another vector orthogonal to

.

Let

be the inner product function. We have the property :

. And I see that

.

Now I must show that

.

.

Wow, it seems I've show part a)! Eventually I must show other properties of

, like that the zero vector is in... and so on.