The linear space C[a,b] of all continuous complex-valued functions on [a,b] with the inner product < f,g > = ò_a^b f(x)[`g(x)] dx is not complete with respect to the norm ||f || = < f,f > ^[1/2] = (ò_a^b |f(x)|² dx)^[1/2]. In fact the sequence (f_n) where

fn(x) = ì ï ï ï ï ï ï ï ï í ï ï ï ï ï ï ï ï î 0 a £ x < b+a 2 (n+n0)(x - b+a 2 ) b+a 2 £ x £ b+a 2 + 1 n+n0 1 b+a 2 + 1 n+n0 < x £ b

(n₀ is a natural number greater than [2/(b-a)])
is a Cauchy but not convergent.