From Euclidean to Hilbert Spaces - Edoardo Provenzi

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orthonormalization process. For a color version of this figure, see www.iste.co.uk/provenzi/spaces.zip

      – Step n, by iteration:

image

      The most important properties of an orthonormal basis are listed in theorem 1.14.

      1) Decomposition theorem on an orthonormal basis:

       [1.7] image

      2) Parseval’s identity7:

      [1.8]image

      3) Plancherel’s theorem8:

      [1.9]image

      Proof of 1: an immediate consequence of Theorem 1.12. Given that (u1, . . . , un) is a basis, v ∈ span(u1, . . . , un); furthermore, (u1, . . . , un) is orthonormal, so image. It is not necessary to divide by ‖ui‖2 when summing since ‖ui‖ = 1 ∀i.

image

      Proof of 3: writing w = v on the left-hand side of Parseval’s identity gives us 〈v, v〉 = ‖v‖2. On the right-hand side, we have:

image

      hence image.

      1) The physical interpretation of Plancherel’s theorem is as follows: the energy of v, measured as the square of the norm, can be decomposed using the sum of the squared moduli of each projection of v on the n directions of the orthonormal basis (u1, ..., un).

      In Fourier theory, the directions of the orthonormal basis are fundamental harmonics (sines and cosines with defined frequencies): this is why Fourier analysis may be referred to as harmonic analysis.

      2) If (u1, . . . , un) is an orthogonal, rather than an orthonormal, basis, then using the projector formula and theorem 1.12, the results of Theorem 1.14 can be written as:

      a) decomposition of v ∈ V on an orthogonal basis:

       [1.10] image

      b) Parseval’s identity for an orthogonal basis:

       [1.11] image

      c) Plancherel’s theorem for an orthogonal basis:

       [1.12] image

      The following exercise is designed to test the reader’s knowledge of the theory of finite-dimensional inner product spaces. The two subsequent exercises explicitly include inner products which are non-Euclidean.

       Exercise 1.1

      Consider the complex Euclidean inner product space

3 and the following three vectors:

image

      1) Determine the orthogonality relationships between vectors u, v, w.

      2) Calculate the norm of u, v, w and the Euclidean distances between them.

      3) Verify that (u, v, w) is a (non-orthogonal) basis of

3.

3 generated by u and w. Calculate PSv, the orthogonal projection of v onto S. Calculate d(v, PSv), that is, the Euclidean distance between v and its projection onto S, and verify that this minimizes the distance between v and the vectors of S (hint: look at the square of the distance).

      5) Using the results of the previous questions, determine an orthogonal basis and an orthonormal basis for

3 without using the Gram-Schmidt orthonormalization process (hint: remember the geometric relationship between the residual vector r and the subspace S).

      6) Given a vector a = (2i, −1, 0), write the decomposition of a and Plancherel’s theorem in relation to the orthonormal basis identified in point 5. Use these results to identify the vector from the orthonormal basis which has the heaviest weight in the decomposition of a (and which gives the best “rough approximation” of a). Use a graphics program to draw the progressive vector sum of a, beginning with the rough approximation and adding finer details supplied by the other vectors.

       Solution to Exercise 1.1

      1) Evidently, image, so by directly calculating the inner products: 〈u, v〉 = −2, 〈u, w〉 = 0 et image.

      2) By direct calculation: image. After calculating the difference vectors, we obtain: image, image.

      3) The three vectors u, v, w are linearly independent, so they form a basis in

3. This basis is not orthogonal since only vectors u and w are orthogonal.

      4) S = span(u, w). Since (u,

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