泛函分析 第6版（影印版）

.ii. applications of the baire-hausdorff theorem
1. the uniform 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 of the condensation of singularities
5. the open mapping theorem
6. the closed graph theorem
7. an application of the closed graph theorem (h6rmander's theorem)
iii. the orthogonal projection and f. riesz' representation theorem
1. the orthogonal projection
2. "nearly orthogonal" elements
3. the ascoli-arzela theorem
4. the orthogonal base. bessel's inequality and parseval's relation
5. e. schmidt's orthogonalization
6. f. riesz' representation theorem
7. the lax-milgram theorem
8. a proof of the lebesgue-nikodym theorem
9. the aronszajn-bergman reproducing kernel
10. the negative norm of p. lax
11. local structures of generalized functions
iv. the hahn-banach theorems
1. the hahn-banach extension theorem in real linear spaces
2. the generalized limit
3. locally convex, complete linear topological spaces
4. the hahn-banach extension theorem in complex linear spaces
5. the hahn-banach extension theorem in normed linear spaces
6. the existence of non-trivial continuous linear functionals
7. topologies of linear maps
8. the embedding of x in its bidual space x"
9. examples of dual spaces
v. strong convergence and weak convergence
1. the weak convergence and the weak* convergence
2. the local sequential weak compactness of reflexive b-
spaces. the uniform convexity
3. dunford's theorem and the gelfand-mazur theorem
4. the weak and strong measurability. pettis' theorem
5. bochner's integral
appendix to chapter v. weak topologies and duality in locally
convex linear topological spaces
1. polar sets
2. barrel spaces
3. semi-reflexivity and reflexivity
4. the eberlein-shmulyan theorem
vi. fourier transform and differential equations
1. the fourier transform of rapidly decreasing functions
2. the fourier transform of tempered distributions
3. convolutions
4. the paley-wiener theorems. the one-sided laplace transform
5. titchmarsh's theorem
6. mikusinski's operational calculus
7. sobolev's lemma
8. garding's inequality
9. friedrichs' theorem
10. the malgrange-ehrenpreis theorem
11. differential operators with uniform strength
12. the hypoellipticity (hormander's theorem)
vii. dual operators
1. dual operators
2. adjoint operators
3. symmetric operators and self-adjoint operators
4. unitary operators. the cayley transform
5. the closed range theorem
viii. resolvent and spectrum
1. the resolvent and spectrum
2. the resolvent equation and spectral radius
3. the mean ergodic theorem
4. ergodic theorems of the hille type concerning pseudoresolvents
5. the mean value of an almost periodic function
6. the resolvent of a dual operator
7. dunford's integral
8. the isolated singularities of a resolvent
ix. analytical theory of semi-groups
1. the semi-group of class (co)
2. the equi-continuous semi-group of class (co) in locally
cofivex spaces. examples of semi-groups
3. the infinitesimal generator of an equi-continuous semigroup of class (co)
4. the resolvent of the infinitesimal generator a
5. examples of infinitesimal generators
6. the exponential of a continuous linear operator whose
powers are equi-continuous
7. the representation and the characterization of equi-con-
tinuous semi-groups of class (co) in terms of the corre-
sponding infinitesimal generators
8. contraction semi-groups and dissipative operators
9. equi-continuous groups of class (co). stone's theorem
10. holomorphic semi-groups
11. fractional powers of closed operators
12. the convergence of semi-groups. the trotter-kato theorem
13. dual semi-groups. phillips' theorem
x. compact operators
1. compact sets in b-spaces
2. compact operators and nuclear operators
3. the rellich-garding theorem
4. schauder's theorem
5. the riesz-schauder theory
6. dirichlet's problem
appendix to chapter x. the nuclear space of a. grothendieck
xi. normed rings and spectral representation
1. maximal ideals of a normed ring
2. the radical. the semi-simplicity
3. the spectral resolution of bounded normal operators
4. the spectral resolution of a unitary operator
5. the resolution of the identity
6. the spectral resolution of a self-adjoint operator
7. real operators and semi-bounded operators. friedrichs'theorem
8. the spectrum of a self-adjoint operator. rayleigh's prin-
ciple,and the krylov-weinstein theorem. the multiplicity
of the spectrum
9. the general expansion theorem. a condition for the
absence of the continuous spectrum
10. the peter-weyl-neumann theorem
11. tannaka's duality theorem for non-commutative compact groups
12. functions of a self-adjoint operator
13. stone's theorem and bochner's theorem
14. a canonical form of a self-adjoint operator with simple spectrum
15. the defect indices of a symmetric operator. the generalized
resolution of the identity
16. the group-ring l1 and wiener's tauberian theorem
xii. other representation theorems in linear spaces
1. extremal points. the krein-milman theorem
2. vector lattices
3. b-lattices and f-lattices
4. a convergence theorem of banach
5. the representation of a vector lattice as point functions
6. the representation of a vector lattice as set functions
xiii. ergodic theory and diffusion theory
1. the markov process with an invariant measure
2. an individual ergodic theorem and its applications
3. the ergodic hypothesis and the h-theorem
4. the ergodic decomposition of a markov process with a
locally compact phase space
5. the brownian motion on a homogeneous riemannian space
6. the generalized laplacian of w. feller
7. an extension of the diffusion operator
8. markov processes and potentials
9. abstract potential operators and semi-groups
xiv. the integration of the equation of evolution
1. integration of diffusion equations in l2(rm)
2. integration of diffusion equations in a compact riemannian space
3. integration of wave equations in a euclidean space rm
4. integration of temporally inhomogeneous equations of
evolution in a b-space
5. the method of tanabe and sobolevski
6. non-linear evolution equations 1 (the komura-kato approach)
7. non-linear evolution equations 2 (the approach through
the crandall-liggett convergence theorem)
supplementary notes
bibliography
index
notation of spaces

.　　are added to the last Chapter XIV of this edition. The author is grateful to Professor Y. KOmura for his careful reading of the manuscript.
　　 Tokyo, April 1974
　　 KOSAKU YOSIDA
　　 Preface to the Fifth Edition
　　 Taking advantage of this opportunity, supplementary notes are added at the end of this new edition and additional references to books have been entered in the bibliography. The notes are divided into two categories. The first category comprises two topics: the one is concerned with
　　the time reversibility of Markov processes with invariant measures, and the other is concerned with the uniqueness of the solution of time dependent linear evolution equations. The second category comprises those lists of recently published books dealing respectively with Sobolev Spaces,Trace Operators or Generalized Boundary Values, Distributions and Hyperfunctions, Contraction Operators in Hilbert Spaces, Choquet's Refinement of the Krein-Milman Theorem and Linear as well as Nonlinear Evolution Equations.
　　 A number of minor errors and a serious one on page 459 in the fourth edition have been corrected. The author wishes to thank many friends who kindly brought these errors to his attention.
　　 Kamakura, August 1077 KOSAKU YOSIDA
　　 Preface to the Sixth Edition
　　 Two major changes are made to this edition. The first is the rewriting of the Chapter VI,6 to give a simplified presentation of Mikusinski's Operational Calculus in such a way that this presentation does not appeal to Titchmarsh's theorem. The second is the rewriting of the
　　Lemma together with its Proof in the Chapter XII,5 concerning the Representation of Vector Lattices. This rewriting is motivated by a letter of Professor E. Coimbra of Universidad Nova de Lisboa kindly suggesting the author's careless phrasing in the above Lemma of the preceding edition.
　　 A number of misprints in the fifth edition have been corrected thanks to kind remarks of many friends.
　　 Kamakura, June 1980 KOSAKU YOSIDA