By A. N. Kolmogorov

Reprinted the following in a single quantity, the 1st half is dedicated to metric and basic areas. starting with a quick creation to set idea and mappings, the authors supply a transparent presentation of the idea of metric and whole metric areas. the primary of contraction mappings and its purposes to the evidence of life theorems within the concept of differential and essential equations gets specified research, as do non-stop curves in metric areas — a subject seldom mentioned in textbooks.

Part One additionally comprises discussions of different matters, equivalent to components of the idea of normed linear areas, susceptible sequential convergence of parts and linear functionals, adjoint operators, and linear operator equations. half specializes in an exposition of degree concept, the Lebesque period and Hilbert area. either elements characteristic quite a few workouts on the finish of every part and contain necessary lists of symbols, definitions, and theorems.

**Read or Download Elements of the Theory of Functions and Functional Analysis. Volume 2: Measure. The Lebesgue Integral. Hilbert Space PDF**

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**Additional resources for Elements of the Theory of Functions and Functional Analysis. Volume 2: Measure. The Lebesgue Integral. Hilbert Space**

**Sample text**

2n). 11) holds for each ν, k = 0, 1, . . , 2n. Therefore, taking 2n + 1 arbitrary values f0 , f1 , . . , f2n , we can express the polynomial Φ(x), which satisfies the conditions Φ(xk ) = fk (k = 0, 1, . . 12) in the form 2n Φ(x) = ∗ k (x)fk . , fk ≡ f (xk ) (k = 0, 1, . . , 2n), we say that Φ(x) is the trigonometric interpolation polynomial of f and we denote it by L∗n f (x) or L∗n (f ; x). 13) is known as the trigonometric interpolation formula (in the Lagrange form). 5). 6). In fact, setting 2n u(x) := sin ν=0 x − xν 2 and uk (x) := u(x) x − xk , sin 2 we have 2n sin ν=0 ν=k xk − xν = uk (xk ) = 2u (xk ) , 2 and then we obtain ∗ k (x) = u(x) uk (x) = .

8) If xk (k = 0, 1, . . , 2n) are distinct points in T, for example 0 ≤ x0 < x1 < · · · < x2n < 2π, then, we have to evaluate the determinant 1 cos x0 sin x0 cos 2x0 sin 2x0 · · · cos nx0 sin nx0 1 cos x1 sin x1 cos 2x1 sin 2x1 cos nx1 sin nx1 . D = .. 3 Chebyshev Systems and Interpolation 41 At first, we transform its elements to the complex form. Multiplying the 3rd, 5th, . . columns by the imaginary unit i and then adding them to the 2nd, 4th, . . columns, respectively, we get 1 eix0 sin x0 ei2x0 sin 2x0 · · · einx0 sin nx0 1 eix1 sin x1 ei2x1 sin 2x1 einx1 sin nx1 .

2! · · · (n − 1)! = 0. This means that the Taylor polynomial exists and is unique. 4 In the case Xn = Pn−1 , φk (x) = x k−1 (k = 1, . . , n), A = [a, b], and a ≤ x1 < · · · < xm ≤ b equipped with multiplicities kj > 0 (j = 1, . . 26) is given by m kj −1 j =1 ν=0 det G = (xk − xj ) = 0. ν! j