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From Euclidean to Hilbert Spaces. Edoardo Provenzi
Читать онлайн.Название From Euclidean to Hilbert Spaces
Год выпуска 0
isbn 9781119851301
Автор произведения Edoardo Provenzi
Жанр Математика
Издательство John Wiley & Sons Limited
Note, too, that solving a linear system of n equations with n unknown variables generally involves far more operations than the calculation of inner products; this highlights one advantage of having an orthogonal basis for a vector space.
THEOREM 1.11.– Let B = {u1, . . . , un} be an orthogonal basis of (V, 〈, 〉). Then:
Notably, if B is an orthonormal basis, then:
PROOF.– B is a basis, so there exists a set of scalars α1, . . . , αn such that
so
Geometric interpretation of the theorem: The theorem that we are about to demonstrate is the generalization of the decomposition theorem of a vector in plane ℝ2 or in space ℝ3 on a canonical basis of unit vectors on axes. To simplify this, consider the case of ℝ2.
If
which is a particular case of the theorem above.
We will see that the Fourier series can be viewed as a further generalization of the decomposition theorem on an orthogonal or orthonormal basis.
1.6. Orthogonal projection in inner product spaces
The definition of orthogonal projection can be extended by examining the geometric and algebraic properties of this operation in ℝ2 and ℝ3. Let us begin with ℝ2.
In the Euclidean space ℝ2, the inner product of a vector v and a unit vector evidently gives us the orthogonal projection of v in the direction defined by this vector, as shown in Figure 1.2 with an orthogonal projection along the x axis.
The properties verified by this projection are as follows:
1) projecting onto the x axis a second time, vector Pxv obviously remains unchanged given that it is already on the x axis, i.e.
2) the difference vector between v and its projection v − Pxv is orthogonal to the x axis, as we see from Figure 1.3;
Figure 1.2. Orthogonal projection
Figure 1.3. Visualization of property 2 in ℝ2. For a color version of this figure, see www.iste.co.uk/provenzi/spaces.zip
3) Pxv minimizes the distance between the terminal point of v and the x axis. In Figure 1.2,
We wish to define an orthogonal projection operation for an abstract inner product space of dimension n which retains these same geometric properties.
Analyzing orthogonal projections in ℝ3 helps us to establish an idea of the algebraic definition of this operation. Figure 1.4 shows a vector v ∈ ℝ3 and the plane produced by the orthogonal vectors u1 and u2. We see that the projection p of v onto this plane is the vector sum of the orthogonal projections
Figure 1.4. Orthogonal projection p of a vector in ℝ3 onto the plane produced by two unit vectors. For a color version of this figure, see www.iste.co.uk/provenzi/spaces.zip
Generalization should now be straightforward: