Question

Let \(V\) be the real vector space of continuous real-valued functions on the closed interval \([0,1]\), and let \(w \in V\). For \(p, q \in V\), define \(\langle p, q\rangle=\int_{0}^{1} p(x) q(x) w(x) d x\).
a) Suppose that \(w(a)>0\) for all \(a \in[0,1]\). Does it follow that the above defines an inner product on \(V\) ? Justify your assertion.

b) Does there exist a choice of \(w\) such that \(w(1 / 2)<0\) and such that the above defines an inner product on \(V ?\) Justify your assertion.