Chapter 5 - Inner Product Spaces - 5.1 Definition of an Inner Product Space - Problems - Page 351: 15

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We are given $f \in V,$ $f(x) =0: x \in [0,\frac{1}{2}]\\ =2x-1: x \in [\frac{1}{2},1]$ We can notice that $f \ne 0 $, but if we take $=\int _0^{\frac{1}{2}}(f(x))^2dx=\int_0^{\frac{1}{2}}0dx=0$ then the property of an inner product $f=0$ is not satisfied. Hence, the given mapping does not define a valid inner product on $V$.