Читать книгу From Euclidean to Hilbert Spaces - Edoardo Provenzi - Страница 17

The “geometric” definition of an inner product in ℝ² and ℝ³ indicates that this product is zero if and only if ϑ, the angle between the vectors, is π/2, which implies cos(ϑ) = 0.

In more complicated vector spaces (e.g. polynomial spaces), or even Euclidean vector spaces of more than three dimensions, it is no longer possible to visualize vectors; their orthogonality must therefore be “axiomatized” via the nullity of their scalar product.

DEFINITION 1.5.– Let (V, 〈, 〉) be a real or complex inner product space of finite dimension n. Let F = {v₁, · · · , v_n} be a family of vectors in V . Thus:

– F is an orthogonal family of vectors if each different vector pair has an inner product of 0: 〈v_i, v_j〉 = 0;

– F is an orthonormal family if it is orthogonal and, furthermore, ‖v_i‖ = 1 ∀i. Thus, if is an orthogonal family, is an orthonormal family.

An orthonormal family (unit and orthogonal vectors) may be characterized as follows:

δ_i,j is the Kronecker delta⁵.