# Are all inner product spaces over the field of the real numbers Hilbert spaces?

All Cauchy sequences over $\mathbf {R}$ converge. Does this mean every inner product space over $\mathbf {R}$ is a complete metric space? If not, what is an example a non-Hilbert inner product space?

• The completion of an inner product space is always well defined and is an Hilbert space. Aug 11 '17 at 5:33

Not every inner product space is a Hilbert space. Every finite-dimensional one is but there are infinite-dimensional ones which aren't complete, say the space of continuous functions on $[0,1]$ with $(f,g)=\int_0^1 f(x)g(x)\,dx$. Some authors call inner-product spaces "pre-Hilbert spaces" for this reason (the completion is a Hilbert space).
• It is worth to note that the completion of the space you mention is $L^2[0,1]$, probably known to anyone who has taken a measure theory course. Aug 11 '17 at 4:59
• I found one: $u_n(x) := max((2x)^n,1)$ Aug 11 '17 at 20:46