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RIESZ SPACES EUR 3140.Θ ÄO!·'· 1 \n\< tl '*·»ΤΗΗΊ)ίβ§"ίί IJM llHi 'li»**. 'tí fe É!>?jpfc EUROPEAN ATOMIC ENERGY COMMUNITY EURATOM LECTURES ON " RIESZ SPACES " given by Prof. A. C. ZAANEN iiil^{TT"1! i ■ BIT' '.'I . * . Ι'βΗΤ Editor : R. F. GLODEN !!i>M!?ÄÉ«il Siffifi 1966 loint Nuclear Research Center Ispra Establishment - Italy Scientific Information Processing Center - CETIS »»cm*if'-WW;flW4)apro 'BfiW»¡kHh;i.u·^: 2Λ*.;;, tf! :1Γ:«Μ$ ■ ■ J'ÎHO *sUr! if nb-. I;·"'-ii ;Γ^*Ε»"^Β1 hiik*MW!?5?'J.-i,K^ fill"; UP» LEGAL NOTICE This document was prepared under the sponsorship of the Commission of the European Atomic Energy Community (EURATOM). Neither the EURATOM Commission, its contractors nor any person acting- on their behalf : Make any warranty or representation, express or implied, with respect to the accuracy, completeness, or usefulness of the information contained in this document, or that the use of any information, apparatus, method, or process disclosed in this document may not infringe privately owned rights ; or Assume any liability with respect to the use of, or for damages resulting from the use of any information, apparatus, method or process disclosed in this document. This report is on sale at the addresses listed on cover page 4 at the price of FF 8.50 FB 85 DM 6.80 Lit. 1060 Fl. 6.20 When ordering, please quote the EUR number and the titl which are indicated on the cover of each report. m ■pejs!« EUR 3140.e LECTURES ON « RIESZ SPACES » given by Prof. A.C. ZAANEN Editor : R.F. GLODEN European Atomic Energy Community — EURATOM Joint Nuclear Research Center — Ispra Establishment (Italy) Scientific Information Processing Center — CETIS Brussels, October 1965 — 66 Pages — 26 Figures — FB 85 This report contains a sequence of lectures on functional analysis given by A.C. ZAANEN, professor at the University of Leyden, invitated by the Euratom Scientific Information Processing Center (Ispra Establishment), during the period from April 4 to April 21, 1966. The present lectures are devoted to the theory of RIESZ spaces (linear vector lattices; lineare Vektorverbände; espaces de RIESZ). Contributions to the theory, founded by F. RIESZ around 1930, are EUR 3140.e LECTURES ON « RIESZ SPACES » given by Prof. A.C. ZAANEN Editor : R.F. GLODEN European Atomic Energy Community — EURATOM Joint Nuclear Research Center — Ispra Establishment (Italy) Scientific Information Processing Center — CETIS Brussels, October 1965 — 66 Pages — 26 Figures — FB 85 This report contains a sequence of lectures on functional analysis given by A.C. ZAANEN, professor at the University of Leyden, invitated by the Euratom Scientific Information Processing Center (Ispra Establishment), during the period from April 4 to April 21, 1966. The present lectures are devoted to the theory of RIESZ spaces (linear vector lattices ; lineare Vektorverbände ; espaces de RIESZ). Contributions to the theory, founded by F. RIESZ around 1930, are EUR 3140.e LECTURES ON « RIESZ SPACES » given by Prof. A.C. ZAANEN Editor : R.F. GLODEN European Atomic Energy Community — EURATOM Joint Nuclear Research Center — Ispra Establishment (Italy) Scientific Information Processing Center — CETIS Brussels, October I960 — 66 Pages — 26 Figures — FB 85 This report contains a sequence of lectures on functional analysis given by A.C. ZAANEN, professor at the University of Leyden, invitated by the Euratom Scientific Information Processing Center (Ispra Establishment), during the period from April 4 to April 21, 1966. The present lectures are devoted to the theory of RIESZ spaces (linear vector lattices ; lineare Vektorverbände ; espaces de RIESZ). Contributions to the theory, founded by F. RIESZ around 1930, are due mainly to Japanese mathematicians (H. NAKANO and his school ; T. OGASAWARA and his school) and Soviet mathematicians (KANTOROVITCH and his school). Recently, new results were ob• tained by W.A.J. LUXEMBURG (Pasadena, USA) and the present lecturer. In the first ten lectures every statement is proved ; in the last two lectures some proofs are only indicated or altogether omitted. due mainly to Japanese mathematicians (H. NAKANO and his school ; T. OGASAWARA and his school) and Soviet mathematicians (KANTOROVITCH and his school). Recently, new results were ob• tained by W.A.J. LUXEMBURG (Pasadena, USA) and the present lecturer. In the first ten lectures every statement is proved ; in the last two lectures some proofs are only indicated or altogether omitted. due mainly to Japanese mathematicians (H. NAKANO and his school ; T. OGASAWARA and his school) and Soviet mathematicians (KANTOROVITCH and his school). Recently, new results were ob• tained by W.A.J. LUXEMBURG (Pasadena, USA) and the present lecturer. In the first ten lectures every statement is proved ; in the last two lectures some proofs are only indicated or altogether omitted. EUR 3140.e EUROPEAN ATOMIC ENERGY COMMUNITY - EURATOM LECTURES ON " RIESZ SPACES " given by Prof. A. C. ZAANEN Editor : R. F. GLODEN 1966 Joint Nuclear Research Center Ispra Establishment - Italy Scientific Information Processing Center - CETIS CONTENTS Lecture 1 : Ordered vector epace, RIESZ space; Positive cone Simple properties of f and f 1 Lecture 2 : Properties of sup (f,g) and inf (f,g)i Standard decomposition of an element; Triangle inequality; Distributive laws · 6 Lecture 3 î More on the distributive laws; BIRKHOFF's inequali­ ties; Disjointness 11 Lecture k i Disjointness; Monotone sequences; Dominated decom­ position theorem; Directed sets 18 Lecture 5 '· Examples of RIESZ spaces 23 Lecture 6 : The ordered vector space of all HERMITIAN operatore on a HILBERT space; RIESZ subspaces, ideals and bands; Examples; Ideal or band generated by a subset ....... 28 Lecture 7 '· Algebraic and direct sums of ideale or bands; Dis­ joint complements J>h Lecture 8 : The BOOLEAN algebra of all disjoint complements; ARCHIMEDEAN RIESZ spaces hO Lecture 9 : DEDEKIND complete and DEDEKIND (T-complete spaces; Projection bands *t6 Lecture 10: Positive and order bounded linear functional6; Order dual of a RIESZ space 31 Lecture 11: DEDEKIND completeness of the order dual; Integrale and singular functionals · 36 Lecture 12: Normed RIESZ spaces 61 SUMMARY This report contains a sequence of lectures on functional analysis given by A.C. ZAANEN, professor at the University of Leyden, invitated by the Euratom Scientific Information Processing Center (Ispra Establishment), during the period from April 4 to April 21, 1966. The present lectures are devoted to the theory of RIESZ spaces (linear vector lattices ; lineare Vektorverbände ; espaces de RIESZ). Contributions to the theory, founded by P. RIESZ around 1930, are due mainly to Japanese mathematicians (H. NAKANO and his school ; T. OGASAWARA and his school) and Soviet mathematicians (KANTOROVITCH and his school). Recently, new results were ob• tained by W.A.J. LUXEMBURG (Pasadena, USA) and the present lecturer. In the first ten lectures every statement is proved ; in the last two lectures some proofs are only indicated or altogether omitted. - 1 - 1 Leoture Definition of an ordered vector space L is called : ordered vector space, if (i) li is a real linear vector space, (ii) L is a partially ordered set, (i.e. for elements f, g, h,... and f ί g, we have f « g and g < h=>f * h f i f for all f f ί g and g « f=*f = g). These two structures are compatible,L is such that a) f « gs$>f+h « g+h for all h ( L, b) f < g=»>af « ag for all real a > 0. Let us recall the following definitions : If L is partially ordered and A is a subset of L, than the element g « L is called an upper bound of A if f c g for all f f A. If, in addition, g « g. for every other upper bound g. of A, then g is called the least upper bound of A, or the supremum of A« Notation : g = sup A or g = sup Γ f : f « A1 Similarly, if h { f for all f β A, then h is a lower bound of A. If h * h. for any other lower bound h. of A, then h is the greatest lower bound of A, or the infimum of A. Notation : h = inf A or h = inf ff : f c A[ Manuscript received on July 29, 1966. - 2 - Definition of a RIESZ space The ordered vector space (o.v. sp.) L is called a RIESZ space if for every pair ?t & € L, there exist sup(f,g) and inf(f,g). Examples (i) The real numbers, i.e. R . o (ii) The plan R . Partial order defined by f1 * «1 t < g4» f2« g2 fig. I, 1 (iii) The spaoe 0([_0,ΐ3)» being the set of all real continuous functions on [0,1] . Definition of f « g : f(x) * g(x) for χ c [0,lj süp(f g) ; \^ fig. 1,2 fig. I, 3 Positive cone of L The positive cone of L, called L*. is by definition I f ; f > 0 0 being the null element of L. Grephio representation of L for the above examples (i) L+ -j ■ 1 1 1 1—I »—< 1 1—·—t- fig. I, if - 3 - Fig. I, 5 (iii) Fig. I, 6 Cone properties t,6 < L+:=*f+g < L+ f * 0=^af * 0 for real a * 0 f, -f e L+=4f = 0 f > 0=4f+g ;> g * 0=^f+g > 0 f > 0Ì =4f = o f < oj Simple properties (i) f > g«f-g e L+ (ii) f)(Hf> sup(f>g) and g = inf(f,g) Λ af ï ag for a > 0 (ui) f>*g* ^ af < ag for a < 0 Indeed f > g<F*f-g * 0<=}-f < -g - k - (iv) sup (f,g) and inf (f,g) : sup (f,g) » - inf (-f, -g) indeed, if h : upper bound of f,g then -h: lower bound of -f, -g.
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