Читать книгу From Euclidean to Hilbert Spaces - Edoardo Provenzi - Страница 14

If (V, 〈, 〉) is an inner product space over , then a norm on V can be defined as follows:

Note that ‖v‖ is well defined since 〈v, v〉 ≽ 0 ∀v ∈ V . Once a norm has been established, it is always possible to define a distance between two vectors v, w in V : d(v, w) = ‖v − w‖.

The vector v ∈ V such that ‖v‖= 1 is known as a unit vector. Every vector v ∈ V can be normalized to produce a unit vector, simply by dividing it by its norm.