An incomplete inner product space.

******************************

The linear space C[a, b] of all continuous complex-valued functions on [a, b] with the inner product < f, g >= ∫_{a}^{b} f(x)\overline{g(x)}dx is not complete with respect to the norm ∥ f ∥=< f, f >^{\frac{1}{2}} = (∫_{a}^{b} |f(x)|^{2}dx)^{\frac{1}{2}} . In fact the sequence (f_{n}) where

f_{n}(x) = \{ \begin{matrix}0 & a ≤ x < \frac{b+a}{2} \\ (n + n_{0})(x − \frac{b+a}{2}) & \frac{b+a}{2} ≤ x ≤ \frac{b+a}{2} + \frac{1}{n+n_{0}} \\ 1 & \frac{b+a}{2} + \frac{1}{n+n_{0}} < x ≤ b \end{matrix}

(n_{0} is a natural number greater than \frac{2}{b−a}) is a Cauchy but not convergent.