From Euclidean to Hilbert Spaces. Edoardo Provenzi

      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

4) Let S be the vector subspace of

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

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

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

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

      4) S = span(u, w). Since (u, w) is an orthogonal basis in S, we can write:

      The residual vector of the projection of v on S is r = v − P_Sv = (2i, 0, 0) and thus d(v, P_Sv)² = ‖r‖² = 4. The most general vector in S is and . This confirms that P_Sv is the vector in S with the minimum distance from v in relation to the Euclidean norm.

      5) r is orthogonal to S, which is generated by u and w, hence (u, w, r) is a set of orthogonal vectors in

³, that is, an orthogonal basis of

³. To obtain an orthonormal basis, we then simply divide each vector by its norm:

      6) Decomposition: .

Plancherel’s theorem: .

      The vector with the heaviest weight in the reconstruction of a is thus r̂: this vector gives the best rough approximation of a. By calculating the vector sum of this rough representation and the other two vectors, we can reconstruct the “fine details” of a, first with ŵ and then with û.

       Exercise 1.2

      Let M(n,

) be the space of n × n complex matrices. The application ϕ : M(n,

) × M(n,

) →

is defined by:

      where denotes the adjoint matrix of B and tr is the matrix trace. Prove that ϕ is an inner product.

       Solution to Exercise 1.2

      The distributive property of matrix multiplication for addition and the linearity of the trace establishes the linearity of ϕ in relation to the first variable.

      Now, let us prove that ϕ is Hermitian. Let A = (ai_,j)_1≼i,j≼n and B = (bi_,j)_1≼i,j≼n be two matrices in M(n,

). Let be the coefficients of the matrix B^† and let be the coefficients of A^†.

      This gives us:

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