Hilbert space

Remarks

Example: $\ell^2$, the space of real sequences $x$ with $\sum_n x_n^2 < \infty$ and $\ip{x}{y} = \sum_n x_n y_n$. The series converges by the Cauchy–Schwarz inequality. It is complete: a Cauchy sequence converges in each coordinate, and the coordinatewise limit is in $\ell^2$ and is the limit in norm.

Non-example: $C([0, 1])$ with $\ip{f}{g} = \int_0^1 f g \, dx$ is an inner product space but not complete. The continuous functions $f_k$ equal to $0$ on $[0, 1/2]$, to $1$ on $[1/2 + 1/k, 1]$ and linear in between form a Cauchy sequence. Their only possible limit is the indicator function of $(1/2, 1]$, which is not continuous.

Every finite-dimensional inner product space is a Hilbert space. An inner product space that is not complete is often called a pre-Hilbert space, and its completion is a Hilbert space.