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Where σ(A) = {λ1 , . . 12). We note that these projections are also functions of A, namely πl = χl (A) where χl (z) = 1, z ∈ B(λl ; δ), 0, z ∈ B(λl ; δ) and δ > 0 is small enough so that σ(A) ∩ B(λl ; δ) = {λl }. 35(ii)). 34. The opposite way is also possible, namely to use functional calculus for ﬁnding the canonical form. 12). , in Dunford & Schwartz [44, Section VII, 1]. 40. Show that sgn det A = (−1)p where p = m(λ) λ∈σ(A) λ<0 for a matrix representation of A with real entries. ) mk 1 Hint.

6. Let X be a connected space, let f : X → Y be continuous. Then f (X) is a connected subset of Y . In particular, if γ : [0, 1] → Y is continuous, A ⊂ Y , and γ(0) ∈ A, γ(1) ∈ A, then there exists t0 ∈ [0, 1] such that γ(t0 ) ∈ ∂A. 7. Let X be a normed linear space and let G be an open subset of X. Then G is connected if and only if for any two points a, b ∈ G there exists a continuous mapping γ : [0, 1] → G such that γ(0) = a, γ(1) = b. In particular, γ can be chosen piecewise linear. Now we are ready to start with the main subject of this section.

J=1 Since n is arbitrary (in an inﬁnite dimensional space) we have obtained the socalled Bessel inequality: ∞ If {en }n=1 is an orthonormal system in X, then ∞ |(y, en )|2 ≤ y 2 for all y ∈ X. 17) n=1 In particular, the sum ∞ |(y, ej )|2 is always convergent. j=1 34 We note that this result, namely the linearity of the operator Pn of the best approximation, is typical for spaces with scalar products. In a general normed linear space X and a ﬁnite dimensional subspace Y the best approximation of an arbitrary x ∈ X by elements of Y exists (by a compactness argument) but a special property of the norm is needed for the uniqueness of the best approximation.

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### Fourier Transform in image processing. And a little Digital Signal Processing

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