Let V be a vector space and < , > be a positive definite inner product. Then show that:

a) < 0, v > = 0 for any v in V

b) Show that for a fixed u in V, < u, v > = 0 for any v in V, then u = 0.

c) Show that < v, w > = 0 for any w in V implies that v = 0.

I did part a) like this:

< 0, v > = < 0 + 0, v > = < 0, v > + < 0, v>

< 0, v > = < 0, v > + < 0, v >

< 0, v > - < 0, v > = < 0, v >

0 = < 0, v>

Now I have been trying to prove parts b) and c) but I cannot think of a way to start. The statements both seem so obvious but I don't know where to begin with a proof.