On the vector space \(\mathcal{F}^{\prime}[a, b]\) of complex continuous
differentiable functions on the, interval \([a, b]\), set
$$
\langle f| g)=\int_{a}^{b} \overline{f^{\prime}(x)} g^{\prime}(x) \mathrm{dr}
\text { where } f^{\prime}=\frac{\mathrm{d} f}{\mathrm{~d} x}, \quad
g^{\prime}=\frac{\mathrm{d} g}{\mathrm{~d} x}
$$
Show that this is not an inner product, but becomes one if restricted to the
space of functions \(f \in\) \(F^{\prime}[a, b]\) having \(f(c)=0\) for seme fixed
\(a \leq c \leq b\). Is it a Hilbert space?
Give a similar analysis for the case \(a=-\infty, b=\infty\), and restricting
functions to those of compact support.