Chapter 5 - Inner Product Spaces - 5.1 Definition of an Inner Product Space - Problems - Page 351: 17

See answer below

We are given $f \in V,$ $f(x) =x, x \in [-1,0]$ We can notice that $f \geq 0 $, but if we take $=\int _{-1}^{0} x[f(x)]^2dx=\int_{-1}^{0}x^3dx=\frac{x^4}{4}| ^0_{-1}=-\frac{1}{4}\lt0$ then the property of an inner product $f=0$ is not satisfied. Hence, the given mapping does not define a valid inner product on $V$.