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Sisalt¨ o¨ Part I: Topological vector spaces 4 1. General topological vector spaces 4 1.1. Vector space topologies 4 1.2. Neighbourhoods and filters 4 1.3. Finite dimensional topological vector spaces 9 2. Locally convex spaces 10 2.1. Seminorms and semiballs 10 2.2. Continuous linear mappings in locally convex spaces 13 3. Generalizations of theorems known from normed spaces 15 3.1. Separation theorems 15 Hahn and Banach theorem 17 Important consequences 18 Hahn-Banach separation theorem 19 4. Metrizability and completeness (Fr`echet spaces) 19 4.1. Baire's category theorem 19 4.2. Complete topological vector spaces 21 4.3. Metrizable locally convex spaces 22 4.4. Fr´echet spaces 23 4.5. Corollaries of Baire 24 5. Constructions of spaces from each other 25 5.1. Introduction 25 5.2. Subspaces 25 5.3. Factor spaces 26 5.4. Product spaces 27 5.5. Direct sums 27 5.6. The completion 27 5.7. Projektive limits 28 5.8. Inductive locally convex limits 28 5.9. Direct inductive limits 29 6. Bounded sets 29 6.1. Bounded sets 29 6.2. Bounded linear mappings 31 7. Duals, dual pairs and dual topologies 31 7.1. Duals 31 7.2. Dual pairs 32 7.3. Weak topologies in dualities 33 7.4. Polars 36 7.5. Compatible topologies 39 7.6. Polar topologies 40 PART II Distributions 42 1. The idea of distributions 42 1.1. Schwartz test functions and distributions 42 2. Schwartz's test function space 43 1 2.1. The spaces C (Ω) and DK 44 1 2 2.2. Schwartz's test function space D(Ω) 45 2.3. D(Ω) as a direct inductive limit 47 2.4. Continuity of linear mappings in the test function space D(Ω) 51 3. Distributions and measures 52 3.1. Distributions and their degrees 52 3.2. Derivatives of distributions 53 3.3. Products of distributions and functions 53 3.4. Topology in the space of distributions 54 3.5. Radon-measures are distributions 55 3.6. The product of a distribution and a function 57 4. The support of a distribution 58 4.1. The support of a distribution 58 4.2. Compact supports 58 4.3. All distributions as derivatives 61 5. Convolution 64 5.1. Convolutions of functions 64 5.2. Convolutions of distributions and functions 65 5.3. Convolution -smoothing = Approximative identity 66 5.4. The convolution of two distributions 67 6. Fourier transforms of tempered distributions 70 6.1. The space S and tempered distributions 70 6.2. The classical Fourier-transform 72 6.3. Fourier-transformations of rapidly decreasing functions 73 6.4. Fourier-transforms of tempered distributions 77 Exercise set 8 79 8.5. 79 8.6. 79 8.7. 79 8.8. 80 8.9. 80 8.10. 81 Exercise set 9 82 9.11. 82 9.12. 82 9.13. 83 9.14. 83 9.15. 84 9.16. 84 Exercise set 10 85 10.17. 85 10.18. 85 10.19. 85 10.20. ( Convolution of functions) 86 10.21. 86 10.22. 86 10.23. (The convolution of a distribution and a function) 86 topological vector space 2010 3 10.24. 86 Exercise set 11 87 11.25. 87 11.26. 87 11.27. 88 11.28. 88 11.29. 89 11.30. 89 11.31. 89 4 Part I: Topological vector spaces 1. General topological vector spaces 1.1. Vector space topologies. Definition 1.1. K is R or C with its standard topology. Vector spaces have coef- ficdients in K. A topological vector space is a vector space E with a topology T , such that + : E × E ! E and(1.1) (1.2) · : K × E ! E are continuous. This kind of topology is E is called a vector space topology. Example 1.2. Examples: - normed spaces, - Lebesgue spaces (0<p<1 included!) 1 - weak topology Remark 1.3. In a topological vector space every translation map E ! E : x 7! x + a is a homeomorphism, so T is translation invariant for a 2 E, we have A 2 T if and only if A + a 2 T : Similarly, T is also homothety invariant: for λ 2 K n f0g, we have A 2 T if and only if λA 2 T : Corollary of transla tion invariance: Corollary 1.4. Between topological vector spaces (E; TE) and (F; TF ) a linear map L : E ! F is continuous at any a 2 E if and only if it is continuous at the origin. Furthermore L i such that en uniformly continuous in the following sense: For each open neighbourhood A of the origin of F there exists an open neighbourhood B of the origin of E such that (x − y) 2 B =) (Lx − Ly) 2 A: Proof. If L is continuous at 0, then x 7! x − a 7! L(x − a) 7! L(x − a) + La = Lx | same as L itself | is continuous at a. Similarly prove the converse; if L is continuous at a then it is also continuous at 0. "Uniformly continuous"is exercise. 1.2. Neighbourhoods and filters. Definition 1.5. Let (X; T ) be a topological space and x 2 X. (1) The set U ⊂ X is a neighbourhood of the point x if x is an interior point of U which means, there exists an open set A 2 T , such, that x 2 A ⊂ U. In particular every open set containing x is a neighbourhood of x. 1 1 2 Draw unit ball in 2-dimensional `2 ! topological vector space 2010 5 (2) The set of all neighbourhoods of a point x is the neighbourhood filter of x: Ux := fU ⊂ E U is a neighbourhood of xg: In particular, in a topological vector space the neighbourhood filter of the 2 origin is called U0. (3) A basis K of a topology T is a subset K ⊂ T such that every open set A 2 T is a unuion of some basis sets: A = S A for some A ⊂ K. Remark 1.6. The classical example of a basis for a topology are the open balls in a metric space. Another example is the product of two topological spaces, where the basic open sets are the finite intersections of Cartesian products of open sets. A ⊂ X is a neighbourhood of x if and only if there exists a basis open set K ⊂ X, such that x 2 K ⊂ A: A ⊂ X is open if and only if it is a neighbourhood of each of its points. In a topological vector space, by translation invariance: 8x 2 E: Ux = x + U0 = fx + A A 2 U0g: Definition 1.7. A neighbourhood basis of a point x 2 E is a set Kx ⊂ Ux of neighbourhoods such that . every neighbourhood contains one of the basic ones: 8A 2 Ux 9K 2 Kx : K ⊂ A; in other words: The neighbourhood filter consists of all basis neighbourhoods and their supsets. Example 1.8. In a normed spaxe (E; k · k) a neighbourhood basis of x is given by all x-centered balls. The same holds in any metric space. Definition 1.9. A ⊂ E is balanced, if and only if αA ⊂ A 8jαj ≤ 1: The balanced hull of A ⊂ E is the smallest balanced set containing A. It exists, since the intersection of arbitrarily (even 1) many balanced sets is balanced. \ bal(A) = fB B is balanced and B ⊃ Ag: S Homework problem: Is bal(A) = jα|≤1 αA ? Definition 1.10. A ⊂ E is absorbing if for all x 2 E there exists a number λ > 0, such that λ < jαj =) x 2 αA: This is equivalent to the claim that every line through the origin fλx λ 2 Kg (x 2 E n f0g) contains an origin-centered segment in A, ie. fαx jαj < g, where > 0. The definition is equivalent to that every origon kautta kulkeva suora fλx λ 2 Kg (x 2 E n f0g) sis¨alt¨a¨a jonkin 0-keskisen v¨alin fαx jαj < g, where > 0. (If K = C, the "segment" looks more like a "disk"'.) 2In some texts F(0). 6 absorboiva absorboiva absorboiva absorboimaton balansoitu ei balansoitu balansoitu balansoitu konveksi konveksi ei konveksi konveksi Kuva 1. Geometry in a vector space A balanced set B ⊂ E is absorbing if and only if [ E = αB α>0 Definition 1.11. A subset A ⊂ E is convex, if and only if it contains all segments between its points: x 2 A; y 2 A; 0 < α < 1 =) αx + (1 − α)y 2 A: The convex hull of a set A ⊂ E is the smallest convex set containing A. It exists, since the intersection of arbitrarily (even 1) many convex sets is convex. So \ co(A) = fB B is convex and B ⊃ Ag: Theorem 1.12. The neighbourhood filter of the origin has the following properties: (1.3) A 2 U0 =) A is absorbing. (1.4) 8A 2 U0 9B 2 U0 : B + B ⊂ A: (1.5) 8A 2 U0 9B 2 U0 : B is balanced and closed and B ⊂ A: Proof. (1.3): Let x 2 E. By continuity if the product and 0x = 0, there exists an interval B = BK(0;") = fα 2 K jαj < "g around the origin 0 2 K such, that Bx ⊂ A: (1.4): Addition + : E × E ! E is continuous and +(0; 0) = 0 + 0 = 0. Therefore there exists a standard basis neighbourhood of (0; 0), like W = C × D such, that C + D = +(C × D) ⊂ A Choose B = C \ D.
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