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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.
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