Verifying an Inner Product for Continuous Functions

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Suppose f(x) and g(x) are continuous real-valued functions defined for [0,1]. Define vectors in n, F= ( f(x1), f(x2), ...,f(xn)) and G= g(x1), g(x2), ...,g(xn)), where xk = k/n. Why is <F,G>n = 1/n  f(xk) g(xk) dx not an inner product for the space of continuous functions?

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An inner product for continuous functions is verified. The solution is concise and well-presented.

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