DOCSLIB.ORG

Dality Theory II: Advanced Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Dality Theory II: Advanced Theory DT II c Gabriel Nagy Dality Theory II: Advanced Theory Notes from the Functional Analysis Course (Fall 07 - Spring 08) Besides the weak dual topologies w# introduced in DT I, there exist many other natural topologies on dual pairs. A general recipe for constructing a large class of such topologies is outlined in sub-section A. Subsequent sections contain some important theorems due to (combinations of) Arens, Banach, Grothendieck and Mackey, which are perfect illustrations of the power of the ”Duality Machine.” The entire section is optional, as it abounds in proof-less results (left as exercises). A. The TM-topologies Conventions. Throughout this note K will be one of the fields R or C, and all vector spaces are over K. Whenever we say that “X and Y form a dual pair,” we will understand that one is given two linear injective maps #: X 3 x 7−→ x# ∈ Y0, (1) #: Y 3 y 7−→ y# ∈ X 0, (2) satisfying the equality x#(y) = y#(x), ∀ x ∈ X , y ∈ Y. (3) (See DT I for the interpretation of these maps.) Remarks 1-2. Use the notations as above. # 1. Since the w -topology on X is defined by the family {py}y∈Y , given by py(x) = |y#(x)| = |x#(y)|, by Exercise 1 from from LCVS III, it follows that a non-empty set M ⊂ X is w#-bounded, if and only if, sup |x#(y)| < ∞, ∀ y ∈ Y. (4) x∈M Likewise, a non-empty set M ⊂ Y is w#-bounded, if and only if, sup |y#(x)| < ∞, ∀ x ∈ X . (5) y∈M w# 2. If M ⊂ X is w#-bounded, then so is the closure conv(bal M) . This follows again from Exercise 1 from LCVS III. 1 Exercise 1♥. Show that, if M ⊂ Y is a non-empty w#-bounded set, then its absolute polar M is absorbing (in X ) and furthermore, its associated Minkowski functional qM is given by # qM (x) = sup |y (x)|, ∀ x ∈ X . y∈M Notation. With M as in Exercise 1, when convenient, the Minkowski functional qM will be simply denoted by pM. (When M is a singleton {y}, pM is py, the seminorm used before.) ♥ 1 Exercise 2. Using the notations as above, show that B(pM) = M. Definition. Suppose X and Y are in a dual pairing. Given a non-empty collection M, # consisting of non-empty w -bounded subsets of Y, one constructs the TM-topology as the locally convex topology on X , defined by the family of seminorms QM = {pM : M ∈ M}. In terms of convergence, the topology TM is characterized as follows. For a net (xλ) in TM # X , the condition xλ −−→ 0 is equivalent to the condition: supy∈M |y (xλ)| → 0, ∀ M ∈ M. Likewise, for a non-empty collection N – consisting of non-empty w#-bounded subsets of X – the corresponding topology TN on Y is defined. Exercises 3-4. Given a non-empty collection M of non-empty w#-bounded subsets of S Y, consider the space hMi = span M∈M M . 3♥. Prove that the following are equivalent: (i) TM is Hausdorff; (ii) hMi is w#-dense in Y. ♥ # 4. Prove that if hMi = Y, then TM is stronger than the w -topology (on X ). All locally convex topological vector spaces can be viewed as TM-topologies, as indicated in the following result. Proposition 1. Let (X , T) be a locally convex topological vector space, and form a dual 0 ∗ pair with either Y1 = X (the algebraic dual), or Y2 = X (the topological dual). 1 2 (i) If V is a T-neighborhood of 0 in X , then its absolute polars V in Y1 and V in Y2 1 coincide. In particular, V is contained in Y2. (ii) Every set in the collection 1 MT = {V : V neighborhood of 0 in (X , T)} # # is w -compact (thus w -bounded) in both Y1 and in Y2. (iii) The TMT -topology on X , defined using either pairing (X with Y1, or X with Y2), coincides with T. 1 Using the notations from LCVS III, for a seminorm p on X , we define B(p) = {x ∈ X : p(x) ≤ 1}. 2 0 Proof. (i). By definition, for V a T-neighborhood of 0, its two absolute polars (in Y1 = X ∗ and in Y2 = X ) are given by 0 V 1 = {φ ∈ X : |φ(x)| ≤ 1, ∀ x ∈ V}, ∗ V 2 = {φ ∈ X : |φ(x)| ≤ 1, ∀ x ∈ V}, ∗ 0 so clearly (by the inclusion X ⊂ X ) one has the inclusion V 2 ⊂ V 1 . Therefore, in order ∗ to prove that these two sets coincide, we only need to prove the inclusion V 1 ⊂ X , in other words, that every φ ∈ V 1 is T-continuous. Fix such a φ and let us show that it is continuous at 0. Equivalently this means that, for every ε > 0, the set Nε = {x ∈ X : |φ(x)| ≤ ε} is a T-neighborhood of 0. This is, however, obvious since the condition φ ∈ V 1 clearly yields the inclusion Nε ⊃ εV. # ∗ ∗ (ii). Since the w -topology on Y2 = X coincides with the w -topology, which in turn ∗ 0 2 ∗ the the induced w -topology from Y1 = X , it suffices to show that V is w -compact in ∗ Y2 = X . This, however, follows immediately from the Alaoglu-Bourbaki Theorem. (iii.) We know (see LCVS III, Theorem-Definition 1 and Remark 8), by the definition of the TMT -topology, that the collection W = {εB(pMT ): M ∈ MT, ε > 0} constitutes a fundamental system of neighborhoods of 0 in the TMT -topology. Therefore (by 2 Exercise 2), in order to prove that T coincides with TMT , it suffices to show that : (a) for every M ∈ MT, its absolute polar M is a T-neighborhood of 0, and (b) for every T-neighborhood U of 0, there exists M ∈ MT, such that M ⊂ U. 1 The first assertion is obvious, since every M ∈ MT is of the form M = V , for some T-neighborhood V of 0, so we obviously have the inclusion M = (V 1 ) ⊃ V. To prove (b), we use the fact (see LCVS I) that for every T-neighborhood U of 0, there exists a closed, convex, balanced T-neighborhood of 0, such that V ⊂ U. Using the duality ∗ 2 between X and Y2 = X , and the Absolute Bipolar Theorem, it follows that V = (V )), so if we take M = V 2 , we now have: M = V ⊂ U. Example 1. If one starts with a dual pairing between X and Y, and one considers M to be the collection of all finite subsets of Y, the associated TM-topology on X is precisely the w#-topology. In sub-sections B and C two other choices for M will be used to produce two new topologies. Remark 3. Suppose X and Y are in a dual pairing. Since, for two non-empty collections # M ⊂ N (of non-empty w -bounded subsets of Y), we have the inclusions QM ⊂ QN, it follows that the associated topologies satisfy the same inclusion TM ⊂ TN. Comment. In connection with the above remark, it is natural to ask the following question. Given two collections M and N, when do the topologies TM and TN coincide, or when does one have the inclusion TM ⊂ TN? Of course, we already know that, a sufficient 2 Whether MT is regarded in Y1 or in Y2, its absolute polar in X is defined the same way, so we do not need subscripts. By Exercise 2, we know that B(pM) = M. 3 condition for the inclusion TM ⊂ TN is the inclusion M ⊂ N, but this is clearly not necessary. (For instance, if M is the collection of all finite subsets in Y with at least two elements, and # N is the collection of all singleton sets in Y, then TM and TN coincide with the w -topology on X , but clearly M ∩ N = ∅.) The exercise below is meant to justify the terminology that will help us answer the above questions. Exercise 5♥. Fix X and Y as above, and fix some non-empty w#-bounded subset M ⊂ Y. Prove the following: (i) pαM = |α| · pM, ∀ α ∈ K; # (ii) the absolute bipolar (M ) is also w -bounded in Y, and p(M) = pM; (iii) if N is a non-empty subset of M, then pN ≤ pM. Definitions. Suppose X and Y are in a dual pairing, and M is a non-empty collection of non-empty w#-bounded subsets of Y S # A. We say that M is total, if its support hMi = span M∈M M is dense in Y in the w - topology. By Exercise 3, this condition is equivalent to the fact that the TM-topology on X is Hausdorff. B. We say that M is saturated, if: (i) M ∈ M ⇒ N ∈ M, ∀ N ⊂ M (non-empty); (i) M ∈ M ⇒ tM ∈ M, ∀ t > 0; (iii) M ∈ M ⇒ (M) ∈ M; (iv) M is directed, i.e. for any M1, M2 ∈ M, there exists M ∈ M, such that M ⊃ M1 ∪ M2. C. It is not so hard to show that the intersection of an arbitrary family of saturated collections is again saturated. Therefore, one can construct Msat to be the intersection of all saturated collections that contain M, so that the collection Msat is the smallest saturated collection that contains M. The collection Msat is called the saturation of M. (See Exercise 6 for more clarifications.) ♥ # Exercise 6. Prove that the collection BY , of all non-empty w -bounded subsets in Y, is saturated. Therefore, the collection used in the saturation construction in Definition C above is non-empty.
Load more

Recommended publications
  • View of This, the Following Observation Should Be of Some Interest to Vector Space Pathologists
    Can. J. Math., Vol. XXV, No. 3, 1973, pp. 511-524 INCLUSION THEOREMS FOR Z-SPACES G. BENNETT AND N. J. KALTON 1. Notation and preliminary ideas. A sequence space is a vector subspace of the space co of all real (or complex) sequences. A sequence space E with a locally convex topology r is called a K- space if the inclusion map E —* co is continuous, when co is endowed with the product topology (co = II^Li (R)*). A i£-space E with a Frechet (i.e., complete, metrizable and locally convex) topology is called an FK-space; if the topology is a Banach topology, then E is called a BK-space. The following familiar BK-spaces will be important in the sequel: ni, the space of all bounded sequences; c, the space of all convergent sequences; Co, the space of all null sequences; lp, 1 ^ p < °°, the space of all absolutely ^-summable sequences. We shall also consider the space <j> of all finite sequences and the linear span, Wo, of all sequences of zeros and ones; these provide examples of spaces having no FK-topology. A sequence space is solid (respectively monotone) provided that xy £ E whenever x 6 m (respectively m0) and y £ E. For x £ co, we denote by Pn(x) the sequence (xi, x2, . xn, 0, . .) ; if a i£-space (E, r) has the property that Pn(x) —» x in r for each x G E, then we say that (£, r) is an AK-space. We also write WE = \x Ç £: P„.(x) —>x weakly} and ^ = jx Ç E: Pn{x) —>x in r}.
    [Show full text]
  • Bornologically Isomorphic Representations of Tensor Distributions
    Bornologically isomorphic representations of distributions on manifolds E. Nigsch Thursday 15th November, 2018 Abstract Distributional tensor fields can be regarded as multilinear mappings with distributional values or as (classical) tensor fields with distribu- tional coefficients. We show that the corresponding isomorphisms hold also in the bornological setting. 1 Introduction ′ ′ ′r s ′ Let D (M) := Γc(M, Vol(M)) and Ds (M) := Γc(M, Tr(M) ⊗ Vol(M)) be the strong duals of the space of compactly supported sections of the volume s bundle Vol(M) and of its tensor product with the tensor bundle Tr(M) over a manifold; these are the spaces of scalar and tensor distributions on M as defined in [?, ?]. A property of the space of tensor distributions which is fundamental in distributional geometry is given by the C∞(M)-module isomorphisms ′r ∼ s ′ ∼ r ′ Ds (M) = LC∞(M)(Tr (M), D (M)) = Ts (M) ⊗C∞(M) D (M) (1) (cf. [?, Theorem 3.1.12 and Corollary 3.1.15]) where C∞(M) is the space of smooth functions on M. In[?] a space of Colombeau-type nonlinear generalized tensor fields was constructed. This involved handling smooth functions (in the sense of convenient calculus as developed in [?]) in par- arXiv:1105.1642v1 [math.FA] 9 May 2011 ∞ r ′ ticular on the C (M)-module tensor products Ts (M) ⊗C∞(M) D (M) and Γ(E) ⊗C∞(M) Γ(F ), where Γ(E) denotes the space of smooth sections of a vector bundle E over M. In[?], however, only minor attention was paid to questions of topology on these tensor products.
    [Show full text]
  • The Dual Topology for the Principal and Discrete Series on Semisimple Groupso
    transactions of the american mathematical society Volume 152, December 1970 THE DUAL TOPOLOGY FOR THE PRINCIPAL AND DISCRETE SERIES ON SEMISIMPLE GROUPSO BY RONALD L. LIPSMAN Abstract. For a locally compact group G, the dual space G is the set of unitary equivalence classes of irreducible unitary representations equipped with the hull- kernel topology. We prove three results about G in the case that G is a semisimple Lie group: (1) the irreducible principal series forms a Hausdorff subspace of G; (2) the "discrete series" of square-integrable representations does in fact inherit the discrete topology from G; (3) the topology of the reduced dual Gr, that is the support of the Plancherel measure, is computed explicitly for split-rank 1 groups. 1. Introduction. It has been a decade since Fell introduced the hull-kernel topology on the dual space G of a locally compact group G. Although a great deal has been learned about the set G in many cases, little or no work has been done relating these discoveries to the topology. Recent work of Kazdan indicates that this is an unfortunate oversight. In this paper, we shall give some results for the case of a semisimple Lie group. Let G be a connected semisimple Lie group with finite center. Associated with a minimal parabolic subgroup of G there is a family of representations called the principal series. Our first result states roughly that if we strike out those members of the principal series that are not irreducible (a "thin" collection, actually), then what remains is a Hausdorff subspace of G.
    [Show full text]
  • Appropriate Locally Convex Domains for Differential
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 86, Number 2, October 1982 APPROPRIATE LOCALLYCONVEX DOMAINS FOR DIFFERENTIALCALCULUS RICHARD A. GRAFF AND WOLFGANG M. RUESS Abstract. We make use of Grothendieck's notion of quasinormability to produce a comprehensive class of locally convex spaces within which differential calculus may be developed along the same lines as those employed within the class of Banach spaces and which include the previously known examples of such classes. In addition, we show that there exist Fréchet spaces which do not belong to any possible such class. 0. Introduction. In [2], the first named author introduced a theory of differential calculus in locally convex spaces. This theory differs from previous approaches to the subject in that the theory was an attempt to isolate a class of locally convex spaces to which the usual techniques of Banach space differential calculus could be extended, rather than an attempt to develop a theory of differential calculus for all locally convex spaces. Indeed, the original purpose of the theory was to study the maps which smooth nonlinear partial differential operators induce between Sobolev spaces by investigating the differentiability of these mappings with respect to a weaker (nonnormable) topology on the Sobolev spaces. The class of locally convex spaces thus isolated (the class of Z)-spaces, see Definition 1 below) was shown to include Banach spaces and several types of Schwartz spaces. A natural question to ask is whether there exists an easily-char- acterized class of D-spaces to which both of these classes belong. We answer this question in the affirmative in Theorem 1 below, the proof of which presents a much clearer picture of the nature of the key property of Z)-spaces than the corresponding result [2, Theorem 3.46].
    [Show full text]
  • Sufficient Generalities About Topological Vector Spaces
    (November 28, 2016) Topological vector spaces Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ [This document is http://www.math.umn.edu/~garrett/m/fun/notes 2016-17/tvss.pdf] 1. Banach spaces Ck[a; b] 2. Non-Banach limit C1[a; b] of Banach spaces Ck[a; b] 3. Sufficient notion of topological vector space 4. Unique vectorspace topology on Cn 5. Non-Fr´echet colimit C1 of Cn, quasi-completeness 6. Seminorms and locally convex topologies 7. Quasi-completeness theorem 1. Banach spaces Ck[a; b] We give the vector space Ck[a; b] of k-times continuously differentiable functions on an interval [a; b] a metric which makes it complete. Mere pointwise limits of continuous functions easily fail to be continuous. First recall the standard [1.1] Claim: The set Co(K) of complex-valued continuous functions on a compact set K is complete with o the metric jf − gjCo , with the C -norm jfjCo = supx2K jf(x)j. Proof: This is a typical three-epsilon argument. To show that a Cauchy sequence ffig of continuous functions has a pointwise limit which is a continuous function, first argue that fi has a pointwise limit at every x 2 K. Given " > 0, choose N large enough such that jfi − fjj < " for all i; j ≥ N. Then jfi(x) − fj(x)j < " for any x in K. Thus, the sequence of values fi(x) is a Cauchy sequence of complex numbers, so has a limit 0 0 f(x). Further, given " > 0 choose j ≥ N sufficiently large such that jfj(x) − f(x)j < " .
    [Show full text]
  • Path Distributions on Sequence Spaces
    PATH DISTRIBUTIONS ON SEQUENCE SPACES ERIK G.F. THOMAS Abstract We define the rank of a summabIe distribution and show that finite dimen­ sional Fresnel distributions eiS are summabIe distributions of rank 2, indepen­ dently of the dimension of the space. This motivates the introduction of path distributions, which we discuss in the case of spaces of sequences (discrete time paths). For a continuous quadratic form (Kç,Ç) on a nuclear sequence space E there exists a unique path distribution FK on the dual sequence space E' such that ( e-i(x,~) FK(dx) = e i (K{ ,{}/2, ç E E. lEI If K(i,j) = min(i,j) this gives a discrete-time analogue of a Feynman integral. 1 Introd uction One of the most interesting problems in functional analysis is the appropriate math­ ematical definition of the Feynman path integral and related objects. Although it was not at the time (1948) obvious, the Feynman path integral [11] is an analogue of the Wiener measure on the space G[O, T] of continuous functions, in which the Gaussian marginals Gt" ... ,tn of the Wiener measure for the time instanees 0 < tI < t2 < ... , tn ~ T are replaced by the Fresnel distributions Ftlo .. ,tn = Git1, .. ,itn · The relation between the Wiener measure and the diffusion equation thus becomes a relation between the Feynman integral and the Schrödinger equation. In spite of considerable work on the matter, such as among ot hers [1], [5], [7], [8], [4], [18], [20], [21], [22], [34], the problem remains to define the projective limit F of the marginal distributions Ft, ,..
    [Show full text]
  • The Role of Minimal Idempotents in the Representation Theory of Locally Compact Groups
    Proceedings of the Edinburgh Mathematical Society (1980), 23, 229-238 © THE ROLE OF MINIMAL IDEMPOTENTS IN THE REPRESENTATION THEORY OF LOCALLY COMPACT GROUPS by BRUCE A. BARNES (Received 16th May 1979) 1. Introduction In the representation theory of finite groups, the minimal idempotents of the group algebra play a central role. In this case the minimal idempotents determine irreducible modules over the group algebra, which in turn are in direct correspondence with the irreducible matrix representations of the group; see Chapter IV of the book of C. Curtis and I. Reiner (2). Many of the same ideas generalise to the situation where the group is compact. In addition, minimal idempotents are involved in some important parts of the theory of Hilbert algebras; see M. RieffePs paper (20). For an arbitrary locally compact (LC) group G there has been no exposition concerning the role that minimal idempotents of V(G) or of C*(G) (the group C*-algebraof G)play in the representation theory of G. In fact, it turns out that minimal idempotents of these algebras are intimately related to important (but special) facets of the representation theory of G: square integrable representations, integral representations, and isolated points of G, the dual of G. In this paper we explore the part that minimal idempotents play in the representation theory of a general LC group. A minimal idempotent of C*(G) determines an irreducible (strongly continuous unitary) representation of G. The equivalence class of this represen- tation is an isolated point in G. At least when G is cr-compact, the converse is also true, an isolated point of G corresponds to an irreducible representation of G determined by a minimal idempotent of C*(G) [Proposition 2].
    [Show full text]
  • Topological Vector Spaces II
    Topological Vector Spaces II Maria Infusino University of Konstanz Winter Semester 2017-2018 Contents 1 Special classes of topological vector spaces1 1.1 Metrizable topological vector spaces ............... 1 1.2 Fr´echet spaces............................ 8 1.3 Inductive topologies and LF-spaces................ 14 1.4 Projective topologies and examples of projective limits . 22 1.5 Approximation procedures in spaces of functions . 25 2 Bounded subsets of topological vector spaces 33 2.1 Preliminaries on compactness................... 33 2.2 Bounded subsets: definition and general properties . 35 2.3 Bounded subsets of special classes of t.v.s............. 39 3 Topologies on the dual space of a t.v.s. 43 3.1 The polar of a subset of a t.v.s................... 43 3.2 Polar topologies on the topological dual of a t.v.s. 44 3.3 The polar of a neighbourhood in a locally convex t.v.s. 50 i 4 Tensor products of t.v.s. 55 4.1 Tensor product of vector spaces.................. 55 4.2 Topologies on the tensor product of locally convex t.v.s. 60 4.2.1 π−topology......................... 60 4.2.2 Tensor product t.v.s. and bilinear forms......... 65 4.2.3 "−topology......................... 68 Bibliography 73 Contents The primary source for these notes is [5]. However, especially in the pre- sentation of Section 1.3, 1.4 and 3.3, we also followed [4] and [3] are. The references to results from TVS-I (SS2017) appear in the following according to the enumeration used in [2]. iii Chapter 1 Special classes of topological vector spaces In these notes we consider vector spaces over the field K of real or complex numbers given the usual euclidean topology defined by means of the modulus.
    [Show full text]
  • Arxiv:2106.10011V1 [Math.FA]
    MEAN ERGODIC COMPOSITION OPERATORS ON SPACES OF SMOOTH FUNCTIONS AND DISTRIBUTIONS THOMAS KALMES AND DANIEL SANTACREU ABSTRACT. We investigate (uniform) mean ergodicity of (weighted) composition operators on the space of smooth functions and the space of distributions, respectively, both over an open subset of the real line. Among other things, we prove that a composition operator with a real analytic diffeomorphic symbol is mean ergodic on the space of distributions if and only if it is periodic (with period 2). Our results are based on a characterization of mean ergodicity in terms of Cesàro boundedness and a growth property of the orbits for operators on Montel spaces which is of independent interest. Keywords: Mean ergodic operator; Uniformly mean ergodic operator; Weighted composition operator; Spaces of smooth functions; Spaces of distributions MSC 2020: 47B33, 47B38, 47A35 1. INTRODUCTION In this note we contribute to the investigation of the dynamical behaviour of (weighted) composition operators Cw,φ(u) = w · (u ◦ φ) on the space of smooth functions E (X ) on an ′ open subset X ⊆ R as well as on the space of distributions D (X ), where the symbol φ of Cw,φ is a smoothself map of X , respectively a diffeomorphism of X when dealing with D ′(X ), and the weight w is a complex valued smooth function on X . The space of smooth functions E (X ) is endowed with its natural topology of uniform convergence on compact subsets of X of all derivatives up to an arbitrary finite order while D ′(X ) is equipped with its strong dual topology, being the topological dual space of D(X ), the space of test functions on X .
    [Show full text]
  • Dual Topology of Generalized Motion Groups
    To the memory of Majdi Ben Halima DUAL TOPOLOGY OF GENERALIZED MOTION GROUPS AYMEN RAHALI Communicated by Vasile Br^ınz˘anescu Let K be a connected compact Lie group acting on a finite dimensional vector space (V; h; i): We consider the semidirect product G = K nV: There are a dense open subset Λ of the dual vector space V ∗ of V and a subgroup H of K such that for each ` 2 Λ the stability group K` is conjugate to H: Let ΛH := f` 2 0 Λ; K` = Hg and Λ(H;K) := ΛH =K where ` and ` in ΛH are identified if they ∗ are on the same K-orbit in V : We define the subspace (Gb)H;K of the unitary dual Gb of G by (G) := fIndKnV (ρ ⊗ χ ); ρ 2 H; χ 2 V ; ` 2 Λ g: b H;K HnV ` b ` b (H;K) In this paper, we give a precise description of the set (Gb)H;K and we determine z explicitly the topology of the space (g =G)H;K which is formed by all the ad- G missible coadjoint orbits O(µ,`); where µ is the highest weight of ρ 2 Hb and z ` 2 Λ(H;K): Also, we show that the topological spaces (Gb)H;K and (g =G)H;K are homeomorphic. AMS 2010 Subject Classification: 22D10, 22E27, 22E45. Key words: Lie groups, semidirect product, representations, coadjoint orbits. 1. INTRODUCTION Let G be a locally compact group. By the unitary dual Gb of G; we mean the set of all equivalence classes unitary representations of G equipped with the Fell topology (see [6]).
    [Show full text]
  • SPECTRAL MEASURES, BOUNDEDLY A-COMPLETE BOOLEAN ALGEBRAS and APPLICATIONS to OPERATOR THEORY
    transactions of the american mathematical society Volume 304, Number 2, December 1987 SPECTRAL MEASURES, BOUNDEDLY a-COMPLETE BOOLEAN ALGEBRAS AND APPLICATIONS TO OPERATOR THEORY WERNER J. RICKER ABSTRACT. A systematic study is made of spectral measures in locally convex spaces which are countably additive for the topology of uniform convergence on bounded sets, briefly, the bounded convergence topology. Even though this topology is not compatible for the duality with respect to the pointwise con- vergence topology it turns out, somewhat surprisingly, that the corresponding L'-spaces for the spectral measure are isomorphic as vector spaces. This fact, together with I. Kluvanek's notion of closed vector measure (suitably devel- oped in our particular setting) makes it possible to extend to the setting of locally convex spaces a classical result of W. Bade. Namely, it is shown that if B is a Boolean algebra which is complete (with respect to the bounded conver- gence topology) in Bade's sense, then the closed operator algebras generated by B with respect to the bounded convergence topology and the pointwise convergence topology coincide. 1. Introduction. Recently, the theory of integration with respect to spectral measures has proved to be very successful in extending many classical results, due to W. Bade and N. Dunford in the 1950s, concerned with commutative operator algebras generated by Boolean algebras of projections in Banach spaces [4, Chapter XVII], a priori having nothing to do with nonliability, to the setting of locally convex spaces; see [2, 3, 11 and 17], for example. It is the multiplicativity of spectral measures which causes the theory of integration with respect to such measures to exhibit additional features not found in the theory for arbitrary vector measures.
    [Show full text]
  • Exercises 04
    (February 8, 2013) Functional analysis exercises 04 Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ [This document is http://www.math.umn.edu/~garrett/m/fun/exercises 2012-13/fun-ex-02-08-2013.pdf] Due Fri, 24 Feb 2013, preferably as PDF emailed to me. [04.1] Let λ be a non-zero not-necessarily-continuous linear functional on a topological vector space V . Show that ker λ is dense if and only if λ is not continuous. [04.2] By considering the poles and residues of the meromorphic family of distributions s 2 us = (integration-against) jxj · log jxj (with x 2 R ) 2 @2 @2 on R , show that up to a constant log jxj is a fundamental solution for the Laplacian ∆ = 2 + 2 , that is, @x1 @x2 ∆ log jxj = δ · (constant) (on R2) 2 2 2 [04.3] On the two-torus T = R =Z , let δ1 be the Dirac delta at a point (x1; y1), and δ2 the Dirac delta @2 @2 at another point (x2; y2). Let ∆ = @x2 + @y2 . Determine the Fourier series for a solution u of ∆u = δ1 − δ2, and observe that this Fourier series is not absolutely convergent pointwise, although it is in H1−"(T2) for every " > 0. −∞ 2 Show that the similar equation ∆u = δ1 has no solution in H (T ). Further suppose both points have irrational slope, in the sense that x1=y1 and x2=y2 are not rational. Show 2 +1 1 that u restricted to rational lines Lξ = f(t; tξ): t 2 Rg=Z (with ξ 2 Q) has a Fourier series in H (T ).
    [Show full text]