We define the "inner product" as $\langle (a,b),(c,d)\rangle = ac-bd$ on $R^2$
We want to verify if this is an inner product space. I'm saying it's not because it does not satisfy the positivity property of inner products, that is:
An inner product space $\langle x,x\rangle > 0$ if $x\not=0$.
I tried to prove this by letting $x=(a,b)=(c,d)=(5,5)$ so that we have the inner product $\langle x,x\rangle = \langle (5,5),(5,5)\rangle=5\times 5-5\times 5 = 0$ contradicting the positivity property of inner products.
Does this proof make sense?