Proceedings of Functional Analysis Symposium. Potchefstroom 1974

91 Pages·1974·1.556 MB·English/Afrikaans

by  A.C. Zaanen (ed)

FUNKSIONAALANALISE SIMPOSIUM
FUNCTIONAL ANALYSIS SYMPOSIUM
Potchefstroom 1974

FUNKSIUNAALANALISE
Voordragte gelewer tydens 'n simposium aangebied deur die
Departement Wiskunde en Toegepaste Wiskunde
van die
P.U. vir C.H.D.
in samewerking met die
Suid-Afrikaanse Wiskundevereniging
22, 23 Julie 1974
FUNCTIONAL ANALYSIS
Lectures given at a symposium presented by the
Department of Mathematics and Applied Mathematics
of the
P.U. for C.H.E.
in collaboration with the
South African Mathematical Society
22. 23 July 1974

VDDRWDORD
Hierdie publikasie spruit voort uit die simposium oor Funksionaal-
analise aangebied aan die Potchefstroomse Universiteit vir C.H.U.
in samewerking met die Suid-Afrikaanse Wiskundevereniging. Dit
bied 'n beeld van die verrigtinge van die simposium deur middel
van geskrawe weergawes van al die vnordragte wat hier gelewer is.
Ons waardering gaan aan die WNNR vir die inisiatief om besoeke
van buitelandse w15kundiges te organiseer en grootliks 00k ta
finansier. Langs dié weg kon prof. A.C. Zaanan uit Nederland in
1974 Suid-Afrikaanse universiteite besoek. Up die simposium was
hy ons hoofreferent. Ons bedank hom graag vir sy oriéntaranda
en insiggewende voordragte. an oak vir die waardevolle same-
sprekings wat met hom gevoer kon word.
Verder is ons dank verskuldig aan almal wat voordragte tydens die
simposium gelewar hat. asook aan hulle wat deur hulle taanwoordig-
haid meegewerk het om daarvan 'n sukses te maak.
Aan die owerheid van die P.U. vir C.H.U. rig ons ’n woord van
hartlike dank en waardering vir die finansiéle en ander steun wat
die aanbieding van die simposium moontlik gemaak hat, sowel as
die galeentheid om die simposiumgangers gesellig te kon onthaal.
W.P. RDBBERTSE
Hoof, Departement Wiskunde en Toegepaste Wiskunde
P.U. vir C.H.O.

INHUUDSUPGAWE/CDNTENTS
A.C. ZAANEN: Riesz spaces and normed Kathe spaces
W. KOTZE: Function algebras with maximal ideal space a
Jordan curve or are ..... ..... 22
J.C. ENGELBRECHT: Die orde van die Fredholm-determinant
vir integraaloperatore in Banach-funksieruimtes 23
G.R.A. GILMDUR: Zero-set spaces and function rings 33
N.J.H. HEIDEMAN: Lipschitz spaces of functions on the
surface of the unit sphere in Euclidean n-space 39
P. VAN ELDIK: Enkele eienskappe van integraaloperatore
in Banach-funksieruimtes ..... 45
I. BRDERE: Digte deelversamelings van parsieel-geordende
versamelings ..... ..... ..... SB
J. SWART: Sekere aspekte van veralgemeende nukleére
ruimtes ... ..... ..... ..... ..... ..... 64
J.H. WEBB: Extended Schauder decompositions of locally
convex spaces u... n-ua- nun-- 71
J.J. GRUBLER: Rate of convergence of certain null
sequences in Banach function spaces 73

RIESZ SPACES AND NURNED KUTHE SPACES
A.C. Zaanen
1. The non-empty set L[elements f. g. h....] is called a
Riesz space if
(1] L is a real vector space
[ii] L is partially ordered (notation f :_g] such that the
vector space structure and the order structure are compatible
(1.9., f: gimpliesf+hig+hfor all heL and afiag
for all real constants a :_O).
[111) the order structure is a lattice structure (1.9., the
least upper bound supr,g] and the greatest lower bound inftf,g]
exist for all f,gé L].
Simple standard examples: [1] L = R. the space of all real
numbers with the usual ordering, [11) L is the space C([a,b]) of
all real continuous functions on the closed interval [a,b] ; f i g
means that +‘(xJ : g[x] for all xe[a,b].
Returning to the general case, we shall denote the zero-element of
L by 0 [there will be no confusion with the number 0);
+
L = (f: f 3.0] is called the positive cone of L; elements in L+
are called positive elements. Furthennore we introduce the
following abbreviations:
f+ = sup[F.UJ; f_ = supI-f.0): Ifl = sup(f,-f).
The proof of the following formulas is immediate.
Addition formula. sup(F + h, g + h) = sup(f.g) * h.
Multiplication formula. suptaf, ag] = a suptf,g] For a > 0.
The same fonnulas hold with sup replaced by inf. Note further-
more that inf(F. g] = -sup (-f,-g] holds for all f, g5 L. By
means of these simple formulas we prove the following theorem.
Theorem 1. Ebr every fé-L we have f = fi - fr, Ifl = f++fr
and inf(f+, f_) : 0. Any element of L is, therefore, the
difference of two positive elements (this is sometimes expressed by
saying that the positive cone is generating).
Proof. we have r + f_ = r + sup[-f, o] = sop(o,+1 = r+. so
F = f+-f—. Furthermore [fl = supIt,~f] = sup(2f.U]-f =
2f+-(£+-f') f++f_. Finally. observing that -f_ = inf(f.DJ,
f'-+'=rlinftr,01=inrtr+f‘.+‘) = inf(+*,+').
we obtain 0
By definition, the elements f and g in L are said to be disjoint
whenever inFI H|.|g|) = 0; this will be denoted by f i g.
+ _
As an example, it was proved in the theorem above that f J.F
holds for every f6 L. For any non-empty subset A of L the