inner product space (original) (raw)

For example, ℝn with the familiar dot product forms an inner product space.

If the metric ∥x-y∥induced by the norm is complete (http://planetmath.org/Complete), then the inner product space is called a Hilbert space .

According to (1), one can define the angle between two non-zero vectors x and y:

cos⁡(x,y):=⟨x,y⟩∥x∥⋅∥y∥.	(2)

This provides that the scalars are the real numbers. In any case, the perpendiculatity of the vectors may be defined with the condition