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Locally Convex Spaces Manv 250067-1, 5 Ects, Summer Term 2017 Sr 10, Fr LOCALLY CONVEX SPACES MANV 250067-1, 5 ECTS, SUMMER TERM 2017 SR 10, FR. 13:15{15:30 EDUARD A. NIGSCH These lecture notes were developed for the topics course locally convex spaces held at the University of Vienna in summer term 2017. Prerequisites consist of general topology and linear algebra. Some background in functional analysis will be helpful but not strictly mandatory. This course aims at an early and thorough development of duality theory for locally convex spaces, which allows for the systematic treatment of the most important classes of locally convex spaces. Further topics will be treated according to available time as well as the interests of the students. Thanks for corrections of some typos go out to Benedict Schinnerl. 1 [git] • 14c91a2 (2017-10-30) LOCALLY CONVEX SPACES 2 Contents 1. Introduction3 2. Topological vector spaces4 3. Locally convex spaces7 4. Completeness 11 5. Bounded sets, normability, metrizability 16 6. Products, subspaces, direct sums and quotients 18 7. Projective and inductive limits 24 8. Finite-dimensional and locally compact TVS 28 9. The theorem of Hahn-Banach 29 10. Dual Pairings 34 11. Polarity 36 12. S-topologies 38 13. The Mackey Topology 41 14. Barrelled spaces 45 15. Bornological Spaces 47 16. Reflexivity 48 17. Montel spaces 50 18. The transpose of a linear map 52 19. Topological tensor products 53 References 66 [git] • 14c91a2 (2017-10-30) LOCALLY CONVEX SPACES 3 1. Introduction These lecture notes are roughly based on the following texts that contain the standard material on locally convex spaces as well as more advanced topics. • J. Horv´ath. Topological vector spaces and distributions. Vol. 1. Reading, Mass.: Addison-Wesley, 1966. isbn: 978-0-486-48850-9 • H. H. Schaefer. Topological Vector Spaces. New York: Springer-Verlag, 1971. isbn: 978-0-387-05380-6 • H. Jarchow. Locally Convex Spaces. Stuttgart: B. G. Teubner, 1981. isbn: 978-3-519-02224-4 • A. Grothendieck. Topological vector spaces. 3rd ed. Philadelphia, PA: Gordon and Breach Science Publishers, 1992. isbn: 0-677-30020-4/hbk • F. Treves. Topological Vector Spaces, Distributions and Kernels. New York: Aca- demic Press, 1976 • L. Narici and E. Beckenstein. Topological Vector Spaces. 2nd ed. Pure and applied mathematics. Boca Raton: Taylor and Francis Group, 2011. isbn: 978-1-58488- 866-6 • A. P. Robertson and W. Robertson. Topological vector spaces. 2nd ed. Cam- bridge Tracts in Mathematics and Mathematical Physics 53. Cambridge Univer- sity Press, 1973. isbn: 978-0-521-20124-7 • G. K¨othe. Topological vector spaces I. Grundlehren der mathematischen Wis- senschaften 159. New York: Springer-Verlag, 1969. isbn: 978-0-387-04509-2 • G. K¨othe. Topological vector spaces II. Grundlehren der Mathematischen Wis- senschaften 237. New York: Springer-Verlag, 1979. isbn: 978-0-387-90400-9 \The principal motivation behind the general theory is the same as that of Banach himself: namely, a search for general tools which might be ap- plied successfully to functional analysis. [...] These efforts culminated in L. Schwartz' theory of distributions (1945), which could be expressed only in the language of locally convex vector spaces." \Analysts are more interested in the properties of a space than in the way it is defined; hence the idea, first clearly formulated by L. Schwartz, of classifying topological vector spaces according to their behaviour with regard to the validity of the main theorems of functional analysis." [D] [git] • 14c91a2 (2017-10-30) LOCALLY CONVEX SPACES 4 2. Topological vector spaces We set K to be either R or C and D := fλ 2 K j jλj ≤ 1g; N = f1; 2; 3;::: g; N0 = f0; 1; 2;::: g: Definition 2.1. Let E be a vector space over K. A topology T on E is called a linear topology if the mappings (x; y) 7! x + y; E × E ! E (λ, x) 7! λx, K × E ! E are continuous. The pair (E; T ) is called a topological vector space (TVS). We write E instead of (E; T ) if it is clear from the context which topology is used. Examples. Normed spaces are topological vector spaces: N 1. c = f(an)n 2 K : limn!1 an existsg with norm k(an)nk = supn janj, 2. c0 = f(an)n 2 c : an ! 0g with the induced norm, 3. ' = f(an)n 2 c : an 6= 0 only for finitely many ng with the induced norm, 4. C(K) = ff : K ! K continuousg where K is a compact topological space, with norm kfk = supx2K jf(x)j, 5. C0(X) = ff : X ! K continuous j 8" > 0 9K ⊆ X compact : supx2XnK jf(x)j ≤ "g where X is a topological space, with norm kfk = supx2X jf(x)j, p N P p P p 1=p 6. l = f(an)n 2 K : n janj < 1g with norm k(an)nkp = ( n janj ) (for 1 ≤ p < 1), 1 N 7. l = f(an)n 2 K : supn janj < 1g with norm k(an)nk1 = supn janj, p R p 8. L (X; F; µ) = ff : X ! K measurable j X jf(x)j dµ(x) < 1g=∼ where (X; F; µ) is a σ-finite measure space and f ∼ g if f = g a.e., with norm k[f]kp = R p 1=p ( X jf(p)j dµ(x)) (for 1 ≤ p < 1), 9. L1(X; F; µ) = ff : X ! K measurable j 9M > 0 such that jf(x)j < M a.e.g=∼ with norm k[f]k1 = inffM > 0 : jf(x)j < M a.e.g. But there are also non-normable topological vector spaces: 10. C(X) = ff : X ! K continuousg with the coarsest topology such that all restric- tion mappings C(X) ! C(K), f 7! fjK for K ⊆ X compact are continuous, 11. lp and Lp for 0 < p < 1, 12. RR with the product topology. Because for z 2 E and λ 6= 0 the map x 7! z + λx is a homeomorphism with inverse x 7! λ−1(x − z) we have: Proposition 2.2. If U is (a basis of) the filter of neighborhoods of 0 then for all z 2 E and λ 6= 0, fz + λU j U 2 U g is (a basis of) the filter of neighborhoods of z. We will refer to the filter of neighborhoods of 0 as the 0-filter and any basis of this filter will be called a 0-basis. Similarly, a neighborhood of 0 will be called a 0-neighborhood. Corollary 2.3. Let E and F be TVS. A linear map f : E ! F is continuous if and only if it is continuous at 0. Corollary 2.4. If U is a 0-basis of a TVS E then for every nonempty subset A ⊆ E we have \ A¯ = fA + U j U 2 U g: [git] • 14c91a2 (2017-10-30) LOCALLY CONVEX SPACES 5 Proof. Because x − U is a basis of the neighborhood filter of x 2 E we have x 2 A¯ , 8U 2 U :(x − U) \ A 6= ; , 8U 2 U : x 2 A + U. Corollary 2.5. If U is a 0-basis of a TVS E then so is fU¯ j U 2 U g. Proof. Given U 2 U , by continuity of addition there is V 2 U such that V¯ ⊆ V + V ⊆ U. Some things become simpler when we can work with a 0-basis consisting of special sets. Definition 2.6. Let E be a vector space. A subset A ⊆ E is called • balanced (or circled) if λx 2 A for all λ 2 D and x 2 A; • absorbent (or radial) if 8x 2 E 9λ0 > 0: x 2 λA for all λ 2 K with jλj ≥ λ0. Because arbitrary intersections of balanced sets are balanced, for every A ⊆ E there exists a smallest balanced set A˘ containing A, called its balanced hull; clearly A˘ = D · A. ¯ If A is a subset of a TVS E we call the closure A˘ of A˘ the closed balanced hull of A. It is the smallest closed and balanced set containing A because if A ⊆ M and M is closed ¯ and balanced we have A˘ ⊆ M˘ = M and hence A˘ ⊆ M¯ = M by the following result. Proposition 2.7. Let E be a TVS and A ⊆ E balanced. Then A¯ is balanced and if 0 2 int(A) then int(A) is balanced. We denote by int(A) denotes the interior of a set A. Proof. D · A ⊆ A implies D · A¯ ⊆ A¯ by continuity of multiplication. For ρ 2 D n f0g we have ρ int(A) = int(ρA) ⊆ int(A), which gives the second claim. Corollary 2.8. If U is a 0-basis in E then fint(U˘) j U 2 U g, fU˘ j U 2 U g and ¯ fU˘ : U 2 U g are 0-bases in E. ¯ Proof. Let U 2 U and choose (Corollary 2.5) U0 2 U such that U0 ⊆ U. By continuity of multiplication there are λ0 > 0 and V 2 U such that λV ⊆ U0 for jλj ≤ λ0. Set ˘ V0 := λ0V ; then V0 ⊆ U0 and for W 2 U such that W ⊆ λ0V we have ˘ ˘ ˘¯ ¯ int(W ) ⊆ int(W ) ⊆ W ⊆ W ⊆ U0 ⊆ U: Proposition 2.9. In a TVS every 0-neighborhood is absorbent. Proof. Let U be a 0-neighborhood in a TVS E and fix x 2 E. Because λ 7! λx is continuous at 0 there is λ0 > 0 such that λx 2 U for jλj ≤ λ0. Next, we identify the minimal requirements on a filter basis such that it generates a linear topology.
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