Let $(X, \F )$ be a vector space where $\F $ is $\R $ or $\C $. An inner product $\ip{\cdot ,\cdot }: X \times X \to \R $ induces a norm $\norm{\cdot }: X \to \R $ defined by $\norm{x} = \sqrt{\ip{x,x}}$ and metric $d: X \times X \to \R $ defined by $d(x, y) = \norm{x - y}$.
If $(X, d)$ is a complete metric space, we call $((X, \F ), \ip{\cdot ,\cdot })$ a complete inner product space (or Hilbert space).1