Editing Inner product space (section)

===Hilbert space===

The article on [[Hilbert spaces]] has several examples of inner product spaces, wherein the metric induced by the inner product yields a [[complete metric space]]. An example of an inner product space which induces an incomplete metric is the space <math>C([a, b])</math> of continuous complex valued functions <math>f</math> and <math>g</math> on the interval <math>[a, b].</math> The inner product is
<math display=block>\langle f, g \rangle = \int_a^b f(t) \overline{g(t)} \, \mathrm{d}t.</math>
This space is not complete; consider for example, for the interval {{closed-closed|−1, 1}} the sequence of continuous "step" functions, <math>\{ f_k \}_k,</math> defined by:
<math display=block>f_k(t) = \begin{cases} 0 & t \in [-1, 0] \\ 1 & t \in \left[\tfrac{1}{k}, 1\right] \\ kt & t \in \left(0, \tfrac{1}{k}\right) \end{cases}</math>

This sequence is a [[Cauchy sequence]] for the norm induced by the preceding inner product, which does not converge to a {{em|continuous}} function.