The question I'm working with is as follows:
Let $V$ be the vector space of polynomials $f(z) = az + b$ for complex $a, b$. Show that the bracket
$\newcommand{\inp}[1]{\left\langle #1 \right\rangle}$ $\inp{p(z),q(z)} = p(0)\overline{q(0)} + p(2)\overline{q(2)}$
defines an inner product on $V$ .
I understand that an inner product has to satisfy the conditions of conjugate symmetry, linearity and positive definiteness, but could someone help me with how to actually form a proof that these conditions hold?
$V$ is a complex vector space if and only if the following properties holds:
1) Linearity in it first argument $$ \langle a+\lambda b,c \rangle=\langle a,c \rangle+\lambda \langle b,c \rangle\qquad\text{ for any }\ a,b,c\in V\ \text{ and any } \lambda \in \Bbb C $$ 2) Conjugate symmetry $$ \langle a,b \rangle=\overline{\langle b,a \rangle}\qquad \text{ for any }\ a,b\in V $$ 3) Definite positiveness $$ \langle a,a \rangle\ge 0\qquad \text{ for any }\ a\in V $$ Every other property of an inner product follows from these three. Thus to prove that your operation is an inner product you only need to check that the equalities in 1), 2) and the inequality in 3) holds for arbitrary elements of $V$ and $\Bbb C$.