泛函分析(第6版) 目录

Contents
0. Preliminaries
1. Set Theory
2. Topological Spaces
3. Measure Spaces
4. Linear Spaces
I. Semi-nonns
1. Semi-nonns and Locally Convex Linear Topological Spaces
2. Nonns and Quasi-nonns
3. Examples of Normed Linear Spaces
4. Examples of Quasi-nonned Linear Spaces
5. Pre-Hilbert Spaces
6. Continuity of Linear Operators
7. Bounded Sets and Bomologic Spaces
8. Generalized Functions and Generalized Derivatives
9. B-spaces and F-spaces
10. Tbe Completion
11. Factor Spaces of a B-space
12. The Partition of Unity
13. Generalized Functions with Compact Support
14. The Direct Product of Generalized Functions
II. Applications of the Baire-Hausdorff Theorem
1. The Unifonn Boundedness Theorem and the Resonance Theorem
2. The Vitali-Hahn-Saks Theorem
3. The Termwise Differentiability of a Sequence of Generalized Functions
4. The Principle ot the Condensation of Singularities
5. The Open Mapping Theorem
6. The Closed Graph Theorem
7. An Application of the Closed Graph Theorem (Hormander's Theorem)
III. The Orthogonal Projection and F. Riesz Representation Theo-rem
1. The Orthogonal Projection
2. "Nearly Orthogonal" Elements
……
IV. The Hahn-Banach Theorems
V. Strong Convergence and Weak Convergence
VI. Fourier Transform and Differential Equations
VII. Dual Operators
VIII. Resolvent and Spectrum
IX. Analytical Theory of Semi-groups
X Compact Operators
XI. Nonned Rings and Spectral Representation
XII. Other Representation Theorems in Linear Spaces
XIIT. Ergodic Theory and Diffusion Theory
XIV The Integration of the Equation of Evolution
Supplementary Notes
Bibliography
Index
Notation of Spaces

泛函分析(第6版) 内容简介

片断：
0.Preliminaries
Itisthepurposeofthischaptertoexplaincertainnotionsandtheo-
remsusedthroughoutthepresentbook.ThesearerelatedtoSetTheory,
TopologicalSpaces,MeasureSpacesandLinearSpaces.
1.SetTheory
Sets.xXmeansthatxisamemberorelementofthesetX;xX
meansthatxisnotamemberofthesetX.Wedenotethesetcon-
sistingofallxpossessingthepropertyPby{x;P}.Thus{y;y=x}is
theset{x}consistingofasingleelementx.Thevoidsetisthesetwith
nomembers,andwillbedenotedby0.IfeveryelementofasetXisalso
anelementofasetY,thenXissaidtobeasubsetofYandthisfact
willbedenotedbyXY,orYX.Ifisasetwhoseelementsare
setsX,thenthesetofallxsuchthatxXforsomeXXiscalledthe
unionotsetsXin,thisunionwillbedenotedbyUX.Theinter-
sectionofthesetsXinisthesetofallwhichareelementsofevery
XX.;thisintersectionwillbedenotedbyX.Twosetsaredis-
jointiftheirintersectionisvoid.Afamilyofsetsisdisjointifevery
pairofdistinctsetsinthefamilyisdisjoint.Ifasequence{Xn}n=1,2...
00
ofsetsisadisjointfamily,thentheunionXmaybewrittenin
theformofasum
Mappings.Thetermmapping,functionandtransformationwillbe
usedsynonymously.Thesymbolf:X->Ywillmeanthatfisasingle-
valuedfunctionwhosedomainisXandwhoserangeiscontainedinY;
foreveryxX,thefunctionfassignsauniquelydeterminedelement
f(x)=yY.Fortwomappingsf:X->Yandg:Y->Z,wecan
definetheircompositemappinggf:X->Zby(gf)(x)=g(f(x)).The
symbolf(M)denotestheset{f(x),xM}andf(x))iscalledtheimage
ofMunderthemappingf.Thesymbolf-1(N)denotestheset{x;f(x}6N}
andf-1(N)iscalledtheinverseimageofNunderthemapping/.Itis
clearthat
1Yoslds.FimctionalAnlysis
Iff:X->Y,andforeachyf(X)thereisonlyonexXwithf{x)=y,
thenfissaidtohaveaninverse(mapping)ortobeone-to-one.Theinverse
mappingthenhasthedomainf(X)andrangeX;itisdefinedbythe
equationx=f-1(y)=f-1({y}).
ThedomainandtherangeofamappingfwillbedenotedbyD(f)and
R(f).respectively.Thus,iffhasaninversethen
ThefunctionfissaidtomapXontoYiff(X)=YandintoYiff(X)Y.
Thefunctionfissaidtobeanextensionofthefunctiongandgarestrictwn
offifD(f)containsD(g),andf(x)==g(x)forallxinD(g).
Zom'sLemma
Definition.LetPbeasetofelementsa.b,...Supposethereisa
hinaryrelationdefinedbetweencertainpairs(a,b)otelementsofP,
expressedbyaaifaThenPissaidtobepartiaUyordered(orsemi-ordered)bytherelation<
Examples.IfPisthesetofallsubsetsofagivensetX,thentheset
inclusionrelation(AB)givesapartialorderingofP.Thesetofall
complexnumbersz=x iy,w=u iv,...ispartiallyorderedby
definingz-Definition.LetPbeapartiallyorderedsetwithelementsa,b,...
Ifacboundorthesupremumotaandb,andwritec=sup(a,b)oraVb
ThiselementofPisuniqueifitexists.Inasimilarwaywedefinethe
greatestlowerboundortheinfimumofaandb,anddenoteitbyinf(a,6)
oraAb.IfaVbandabexistforeverypair(a,6)inapartially
orderedset.P,Piscalledalattice.
Example.ThetotalityofsubsetsMofafixedsetBisalatticeby
thepartialorderingM1M1M2.