Functional Analysis

泛函分析-1

Banach and Hilbert spaces

Lp spaces; C(X); completeness and the Riesz-Fischer theorem; orthonormal bases; linear functionals; Riesz representation theorem; linear transformations and dual spaces; interpolation of linear operators; Hahn-Banach theorem; open mapping theorem; uniform boundedness (or Banach-Steinhaus) theorem; closed graph theorem. Basic properties of compact operators, Riesz- Fredholm theory, spectrum of compact operators. Basic properties of Fourier series and the Fourier transform; Poission summation formula; convolution.

1，泛函分析的起源与历史，泛函分析与数学和自然科学其它分支的关系。

1， 拓扑空间、度量空间、网、范畴、范畴与函子，态射与同构、对象的分类、图。

2， 满射的性质、直积与直和、函子、自由函子、自然变换、等价、Tychonoff拓扑、准范数、范数、准赋范线性空间、赋范线性空间、商准范数。Hilbert空间，Banach空间，

3， Euclid范数、一致范数、赋范线性空间的直和、Minkowski泛函、准度

量、共轭双线性泛函、内积、Cauchy-Bunyakovskii不等式、准Hilbert空间、拟Hilbert空间、正交、正交系、Bessel不等式，正交化，基， Schauder基、有界算子、同构定理。

等距同构、对偶空间、平移算子、积分算子、核、Volterra算子、微分算子。

3，正交完备化定理，Hilbert空间上的线性泛函。

4，度量空间及其完备性，赋范空间，准赋范线性空间，拓扑空间。

5，紧致性，可数紧致性，拓扑空间与度量空间的完备性。

6，C[a,b]，lp与Lp[a,b]空间的准紧致性判据。

4， 有界算子的拓扑与范畴性质、拓扑同构、范数的等价、弱拓扑等价、算子的矩阵、拓扑余子空间、投影算子、凸泛函与线性泛函，Hahn-Banach定理。

5， Riesz定理、对偶空间， 二次对偶空间、自反空间、对偶空间上单位球的弱紧性。

9，C[a,b]，lp与Lp[a,b]空间上的连续线性泛函。

10，线性算子，赋范算子，对偶算子，一致有界原理。

13， 量子泛函分析概述， 

量子范数、量子赋范线性空间、量子化、富山淳定理、Arveson-Wittstock定理、

11，线性算子的拓扑与纲性质，Baire定理、Banach空间、Hilbert空间。Hilbert与Banach空间的纲，Riesz-Fischer定理。

14，逆算子，可逆性，逆算子的Banach定理。

6， Banach空间上的Weierstrass判别法、连续扩张原理、Banach空间与Hilbert空间的范畴、Riesz-Fischer定理、Gowers定理、Enflo-Read定理、正交补、Riesz定理、Phillips定理、开映射原理、一致有界原理。Banach逆算子定理、闭图像定理、Banach-Steinhaus定理。

7， Banach自伴函子，Banach伴随函子、Banach伴随算子、正合序列、赋范线性空间的完备化、完备化的存在性与唯一性、代数张量积、泛函的张量积、Banach张量积、Hilbert与Banach张量积。张量积的存在性与唯一性。

15，算子的谱与分解，分解的解析性质，非空谱，谱的半径公式。

8， 投影张量积的唯一性、Grothendieck定理、Hilbert张量积、不变测度、保测度映射、Koopman引理、von Neumann遍历定理、Birkhoff遍历定理、紧空间、Kuratowski定理、Milyutin定理、局部紧空间、Alexandroff紧化。

9， Hausdorff\varepsilon-网、完全有界、Riesz定理、等度连续、Arzela定理、Baire测度、正线性泛函、Riesz-Markov定理、网的单调收敛定理、复Baire测度、凸集、具有紧支集的连续函数、紧算子、紧算子的谱的Riesz定理，Fredholm定理。

17，自伴紧算子的Hilbert定理。

18，自伴算子的函数，自伴算子的谱理论。

Schauder定理、Enflo定理、Grothendieck逼近定理、Szankowski反例、Schmidt定理。

10， Hilbert-Schmidt算子、Schatten-von Neumann定理、积分算子，Fredholm算子、

Fredholm算子的指标、指标的乘积性质、Fredholm算子的Fredholm择一定理、第二类积分方程、算子方程、Fredholm定理、摄动下算子的稳定性。

11， 积分方程的Fredholm择一定理、区间、平衡集、拓扑线性空间、局部凸空间、多赋范线性空间、可数赋范空间、准范数的弱算子族、准范数族的等价。

12，多范数空间，弱拓扑。 \lambda-弱准范数族与\lambda-弱拓扑、弱星准范数族、弱^*拓扑、弱^*收敛、\lambda-弱连续、弱^*自伴算子、Banach-Alaoglu定理、Krein-Milman定理、弱^*函子、广义函数、增缓广义函数、具有紧支集的广义函数、正则广义函数、奇异广义函数。

22，作为多范数空间的基本函数空间D(Ω)、E(Ω)、S(R^n)。

13， Dirac的/delta-函数、Sobolev广义导数、广义函数的结构、广义函数的磨光化、算子的正则点与奇点、剩余谱、连续谱、复结合代数、代数的单位元、单位代数、特征标、代数的表示、代数的多项式运算、多项式运算的谱映射法则、子代数、双边理想。

泛函分析-2

1， 商代数、Banach代数、Wiener代数、Banach代数的拓扑同构、Hilbert恒等式、Gelfand-Mazur定理、Banach代数的谱半径、谱半径公式、拟幂零Banach代数、整全纯运算、Gelfand定理、Gelfand变换。

24，函数E(Ω)和S(R^n)的理论。

2， 逼近元、Cohen因式分解定理、Schwartz空间上的Fourier变换、Abel群上的群代数、Abel群上的不变测度、交换群上的卷积。Abel群上的卷积运算、Abel群上的卷积运算的基本性质、广义函数及其运算，正则与奇异广义函数。广义函数的卷积运算。

3，酉算子，Fourier算子，Plancherel定理、Hilbert-Fourier变换、Paley-Wiener定理、Sobolev空间、Sobolev单射定理、正则化、偏微分方程的基本解、

\mathcal{D}_{+}^{/}代数。

26，基本与广义函数的Fourier变换，分布的直积与反演。

4， H^1{\Omega}空间、H_0^1{\Omega}空间、Poincare不等式、Rellich定理、Meyers-Serrin定理、自然拓扑、Cauchy网、完备网、有向准范数族、吸收集、分离超平面定理。

5， Frechet空间、不动点、压缩映射原理、Leray-Schauder-Tychonoff定理、仿射线性映射、映射族的公共不动点、Markov-Kakutani定理、不动点定理在常微分方程初值问题局部解的存在性上的应用、交换紧群上的Haar测度、自举方程、散射振幅相的判断、低密度相关函数的存在性、同调群、Banach空间上的隐映射与逆函数定理。

6， Hilbert伴随算子、伴随方程、Fredholm定理、自伴算子、正规算子、自伴算子的谱的性质、正规算子的谱的性质、Hilbert-Schmidt定理、紧算子的极分解、对合代数、对合同态、Banach代数的基本概念。Banach*-代数、等距同构与等距同态、C*-代数的基本概念。Gelfand-Naimark定理。

7， von Neumann双换位子定理、von Neumann代数的基本概念。堺定理、von Neumann定理、连续泛函运算与正算子，连续泛函运算的谱映射法则、任意有界算子的极分解、算子的比较、自伴算子的结合族、预解。

28，算子值Riemann-Stiltjes积分与算子值Lebesgue积分极及其与谱理论的联系。

8， Calkin定理、弱测度族、Borel函数、Borel泛函运算、谱测度、算子的谱测度、自伴算子的Hilbert谱理论、向量的人为测度、循环算子、Hilbert和。

9， 自伴算子（谱理论的几何形式）、自伴算子的Hellinger定理、混合保测度变换、Baker变换、Halmos-von Neumann定理、Radon测度、Dirac测度、Wendel定理、测度局部化原理、层。

10， Banach代数的正则表示、预解集、预解函数、Stone-Weierstrass定理、交换C*-代数的特征化、Stone-Cech紧化、Gelfand-Naimark-Segal结构。

11， 正规算子谱定理的连续泛函运算形式、算子的绝对值、Fuglede定理、正规算子谱定理的Borel泛函运算形式、谱投影、Weyl-von Neumann定理、Banach代数上的强拓扑与弱拓扑、Banach代数的放大、von Neumann双换位子定理的证明、\sigma-强拓扑、w*-拓扑、\sigma-弱连续泛函运算。

12， von Neumann代数的预对偶、极大交换代数、重度自由算子、正规算子谱定理的重度自由算子形式、原子代数、算子的范围、线性变换的图、闭算子、可闭算子、稠定算子、闭算子的预解集、无界算子的谱。

13， 无界对称算子、无界自伴算子、本质自伴算子、自伴算子的基本判据、无界自伴算子的谱理论、投影值测度、强连续单参数酉群、Stone定理、von Neumann定理、自伴算子的交换性、典型交换关系、Weyl关系。

Functional Analysis

1. Peter D. Lax, Functional Analysis, Wiley-Interscience, 2002.　

1，A.Ya.Helemskii，泛函分析讲义，莫斯科不间断数学教育中心，2004。

2，A.N.Kolmogorov、S.V.Fomin，函数论与泛函分析初步，科学出版社，1989。

3，V.S.Vladimirov，数学物理中的广义函数，科学出版社，1979。

4，A.A.Kirillov、A.D.Gvishiani，泛函分析的理论与问题，科学出版社，1988。

5，Frigyes Riesz、Bela Szokefalvi–Nagy，Functional Analysis，Dover Publications。

Kolmogorov, “Elements of the Theory of Functions and Functional Analysis”

L.V.Kantorovitch, G.P.Akilov “Functional Analysis” 

А.Б.安托涅维奇《泛函分析习题集》高等教育出版社

《泛函分析理论习题解答》克里洛夫

H.Brezis “Analyse Fonctionelle” 

Conway, “A Course in Functional Analysis”
Rudin, “Functional Analysis”

Yoshida, “Functional Analysis”

Lang, “Real and Functional analysis”

I. M. Gelfand, “Generalized Functions” I-V

N.Bourbaki “Topological Vector Space”Chpt. 1-5 

H.H.Schaefer, “Topological Vector Spaces”

J.L. Kelley, I. Namioka, “Linear Topological Spaces”

P.R. Halmos, “A Hilbert Space Problem Book”

I.E. Segal, R.A. Kunze, “Integrals and Operators” 

Dunford,Schwarz, “Linear Operators”I 

S.K. Berberian, “Lectures in Functional Analysis and Operator Theory” 

J.M.Bony, 遍历理论(ergodic theory)的书,”那是真正的测度论”(J.M.Bony).

张恭庆，《泛函分析讲义》（上、下册），北大版
夏道行，《实变函数论与泛函分析》（下册），高教版
夏道行,严绍宗,舒五昌,童裕孙 “泛函分析第二教程” 

刘培德《泛函分析基础》武汉大学出版社

郑维行《实变函数与泛函分析概要》（下册）高等教育出版社

汪林 “泛函分析中的反例” 

夏道行,杨亚立 “拓扑线性空间”  

165《实变函数与泛函分析》郭大钧等编

【习题集与辅导书】

165《泛函分析习题集及解答》(印度)V.K.Krishnan 著

166《函数论与泛函分析初步》柯尔莫哥洛夫著

167《泛函分析疑难分析与解题方法》孙清华，孙昊著

168《泛函分析内容、方法与技巧》孙清华, 侯谦民, 孙昊著

《泛函分析概要》刘斯铁尔尼克、索伯列夫

《泛函分析习题集》安托涅维奇

《泛函分析理论习题解答》克里洛夫

【提高】

169《泛函分析中的反例》汪林著

170《泛函分析新讲》定光桂著

173《泛函分析:理论和应用:theorie et applications》Haim Brezis著

函数论与泛函分析的应用问题

1，复Hilbert空间上的自共轭算子及其在循环向量上的作用，复值函数算子的谱，平方可积性与有限可数可加Borel测度的关系。

2，复可分Hilbert空间上的自共轭算子，作为前述段落中的算子的至多可数的直和。

3，复Hilbert空间的Von Neumann定理。

4，复Hilbert空间上的自共轭算子与被实直线上的复值函数定义的空间上的可测有界函数的乘积算子的酉等价。

5，有界自共轭算子的谱分解。

6，Hilbert空间与复Hilbert空间上的有界线性算子的复合，闭算子，Hilbert空间上的自伴与完全自伴算子，线性闭算子的二次复合的存在性证明。

7，对称算子的自伴性判据。

8，局部凸空间的构造方法，投影与诱导极限，和与直和，局部凸空间上的张量积。

9，拓扑线性空间。

10，向量空间的弱拓扑，线性泛函空间的向量子空间。

11，对称算子的亏指数。

12，Von Neumann公式。

13，对称算子的对称扩张，例子。

14，对称算子的亏指数的等价性的Von Neumann判据。

15，对称算子的谱。

16，Cayley变换。

17，无界自伴算子的谱理论，Stone定理，Tauber定理。

18，二次型，无界算子的收敛，Trotter与Chernov定理。

19，自伴算子的摄动，Friedrichs扩张，Kato不等式，Kalf-Walter-Schmincke-Simon定理，Davies-Faris定理。

20，交换子定理。

21，算子半群，无限生成子的函数。

22，遍历理论，点状与连续流遍历理论。

23，热方程与Schrodinger方程的Feynman公式。

24，Feynman-Kac公式。

25，量子力学的公理系统。

26，Bell不等式与量子力学的经典概率模型的不确定性。

27，量子信息与量子计算概述。

1，M.Reed、B.Simon，Methods of Modern Mathematical Physics，Vol 1，Academic Press，1979。

2，M.Reed、B.Simon，Methods of Modern Mathematical Physics，Vol 2，Academic Press，1975。

3，N.Danford、J.T.Schwartz，Linear Operators，Interscience，1963。

4，Е.В.Dаvies，One-Parameter Semigroups，Academic Press，1980。

5，O.G.Smolyanov、E.T.Shavgulidze，陆径积分，莫斯科大学出版社，1990。

6，R.Alicki、M.Fannes，Quantum Dynamical Systems，Oxford University Press，2001。

7，V.I.Smirnov，高等数学教程，第五卷，物理数学书籍出版社。

8，V.I.Bogachev，高斯测度，科学出版社，1997。

9，O.G.Smolyanov、A.Truman，Bell不等式与量子系统的概率模型，俄罗斯科学院报告，397，1，2002，

Functional analysis -1

Banach and Hilbert spaces

Lp spaces; C(X); completeness and the Riesz-Fischer theorem; orthonormal bases; linear functionals; Riesz representation theorem; linear transformations and dual spaces; interpolation of linear operators; Hahn-Banach theorem; open mapping theorem; uniform boundedness (or Banach-Steinhaus) theorem; closed graph theorem. Basic properties of compact operators, Riesz- Fredholm theory, spectrum of compact operators. Basic properties of Fourier series and the Fourier transform; Poission summation formula; convolution.

1 , The origins and history of functional analysis, functional analysis, relations with other branches of mathematics and the natural sciences.

1 , Topological space, metric space, network, category, categories and functors, morphisms and object classification, isomorphism, graph.

2 , Injective property, direct product and direct sum, free functors, and functors, natural transformations, equivalence,Tychonoff topology, norm, norm, quasi-normed linear spaces, normed linear space and quotient norm. Hilbertspace,Banach space,

3 , Euclid norm, uniform norm, and normed linear space,Minkowski functionals, associate degree

, Conjugated double linear functionals, inner product, Cauchy-Bunyakovskii Inequality, Hilbert Space, the proposed Hilbert Space, orthogonality, orthonormal system, Bessel Inequality, orthogonalization, base, Schauder basis, bounded operator, isomorphism theorems.

Isometry, dual integral operator, nuclear, space, shift operator, Volterra Operators, differential operators.

3 Orthogonal completeness theorems, Hilbert The linear functionals on the space.

4 , A metric space and its completeness, normed spaces, normed linear spaces, topological space.

5 , Compactness, countable compactness, completeness of metric spaces and topological spaces.

6 , C[a,b] , lp Lp[a,b] Space compactness criterion.

4 , A bounded operator topology and category properties, topological isomorphism, equivalence, weak topologies of norm equivalence, operators, topology, minors of the matrix space, projection operator, convex functionals with linear functionals,Hahn-Banach theorem.

5 , Riesz theorem, the dual space, second dual space, reflexive spaces, dual spaces on the unit ball of weak compactness.

9 , C[a,b] , lp Lp[a,b] On the space of continuous linear functionals.

10 , Linear operator, normed operator dual operators, the uniform boundedness principle.

13 , Quantum functional analysis overview

Quantum of norm, quantum, quantum normed linear spaces, Fu Shanchun theorem, Arveson-Wittstock Theorem,

11 , Outline of linear operator topology and properties Baire Theorem, Banach Space, Hilbert Space. Hilbert and Banach space is the goal,Riesz-Fischertheorem.

14 Inverse operator, reversibility, inverse Banach Theorem.

6 , Banach space of Weierstrass criterion, continuous expansion principle,Banach spaces and Hilbert spaces category,Riesz-Fischer theorems,Gowers theorem, theEnflo-Read theorem, orthogonal complement,Riesz theorems,Phillips Theorem and the open mapping theorem and the uniform boundedness principle. Banach inverse operator theorem and closed graph theorem, theBanach-Steinhaus theorem.

7 , Banach self-adjoint functors,Banach adjoint functors, andBanach adjoint operator, exact sequence, completion of a normed linear space, complete the existence and uniqueness of algebraic tensor, tensor products of functional, andBanach Tensor product,Hilbert and Banach tensor. Existence and uniqueness of the tensor product.

15 , Operator of the spectral decomposition, decomposition of analytical nature, non-empty spectrum, spectral RADIUS formulas.

8 , Projection tensor product uniqueness,Grothendieck theorem,Hilbert tensor product, the same measure and measure-preserving mappings,Koopman lemmas,von Neumann Ergodic theorem,Birkhoff ergodic theorem, tight spaces andKuratowskitheorems,Milyutin theorem, locally compact spaces, Alexandroff compactification.

9 , Hausdorff\varepsilon- NET, totally bounded,Riesz theorem, equicontinuous,Arzela theorem,Baire measure, positive linear functionals, Riesz-Markov The monotone convergence theorem, theorem, network complex Baire Measure, convex sets and continuous functions with compact support, compact operator, The spectra of compact operators Riesz Theorem Fredholm Theorem.

17 , Self adjoint compact operators Hilbert Theorem.

18 , The function of self-adjoint operators, spectral theory of self-adjoint operators.

Schauder Theorem, Enflo Theorem, Grothendieck Approximation theorems, Szankowski Anti-cases, Schmidt Theorem.

10 , Hilbert-Schmidt operator,Schatten-von Neumann theorem, the integral operators,Fredholm operator,

Fredholm Operator multiplies the index, index properties, Fredholm Operator Fredholm Alternative theorem, the integral equation of the second kind, operator equations, Fredholm Theorem, stability under perturbations operator.

11 , Integral equations Fredholm alternative theorem, band and balance set, topological vector spaces, in locally convex spaces and normed linear spaces, normed spaces, quasi-norm, the norm of the weak operator equivalent.

12 , Norm space weak topology. \Lambda- the weak norm and \lambda- star norm weak topology or weak, weak ^* topology, weak^* convergence,\ Lambda- weakly continuous, weak ^* self-adjoint operator,Banach-Alaoglu theorem, theKrein-Milmantheorem, the weak ^* Functor, generalized functions, growth of generalized functions, with compactly supported generalized functions, generalized singular functions, generalized functions.

22 , As a basic norm space function space D(Ω) 、 E(Ω) 、 S(R^n) 。

13 , Dirac /Delta- function,Sobolev generalized derivative, the structure of generalized functions, generalized functions, operator of the Polish regular singular points, and the remaining spectrum, spectrum, complex associative algebra, algebraic identity, units, feature algebras, algebraic expressions, Algebraic polynomial operations, polynomial spectral mapping rules of operation, number of children, two sided ideal.

Functional analysis -2

1 , Quotient algebras,Banach algebra, theWiener algebra,Banach algebras of topological isomorphism,Hilbert identities,Gelfand-Mazur Theorem, Banach Spectral RADIUS, spectral RADIUS formulas, Algebra to be nilpotent Banach Algebra, integral holomorphic operations, Gelfand Theorem, Gelfand Transform.

24 , The function E(Ω) S(R^n) Theory.

2 , Approximation, andCohen factorization theorem, theSchwartz space of Fourier transforms,Abel Group on the Group algebra, Abel Group invariant measure on, Exchange Group of convolution. Abel Group convolution operation,Abel convolution operation on the Group’s basic properties, generalized functions and operations, regular and singular generalized functions. Generalized convolution of functions.

3 , Unitary operators, Fourier Operators, Plancherel Theorem, Hilbert-Fourier Transform, Paley-Wiener Theorem, Sobolev Space, SobolevInjective theorem, regularization, basic solutions of partial differential equations,

\mathcal{D}_{+}^{/} Algebra.

26 Basic and generalized function Fourier Transformation and distribution of direct product and inversion.

4 , H^1{\Omega} space,H_0^1{\Omega} space,Poincare inequality,Rellich theorem, Meyers-Serrin theorem, the natural topology,Cauchy nets, sporting nets, quasi-norm, absorbing set, the separating hyperplane theorem.

5 , Frechet space, fixed point, contraction mapping principle, theLeray-Schauder-Tychonoff theorem, Ray-like maps, mapping and family of common fixed point andMarkov-Kakutani theorem, Fixed point theorem on the existence of solutions of initial value problem of partial differential equation application, Exchange on a tight group of Haarmeasure, bootstrap equation and scattering amplitude-phase judgment, the existence of low-density correlation functions, homology groups,Banach space mapping implicit and inverse function theorems.

6 , Hilbert adjoint operator, with equations,Fredholm theorem, self-adjoint operator, the formal nature of the operator, the spectrum of self-adjoint operators, operators of the spectrum of properties,Hilbert-Schmidt the polar decomposition theorem, compact operators, involutive algebra, the contract States, Banach algebra concepts. Banach*-algebra, isometry and isometric homomorphism, andc *- algebraic concepts. Gelfand-Naimark theorem.

7 , Von Neumann double commutator theorem,von Neumann algebra concepts. Sakai theorem,von Neumann theorem, continuous functional operationsand operators, continuous spectral mapping of functional operation rules, any bounded operator polar decomposition, operator of the comparison, the combination of self-adjoint operators, the resolvent.

28 , Operator-valued Riemann-Stiltjes Integral and operator value Lebesgue Integral and its link with spectral theory.

8 , Calkin theorem, a weak measure,Borel function,Borel functional operation, operator, spectrum measurement of the spectral measure, self adjoint operators in Hilbert spectral theory, vectors of human measure, cycle operator, Hilbertand.

9 , Self-adjoint operators (spectral theory of geometric forms), self-adjoint operators Hellinger theorem, mix measure-preserving transformations,Baker transformation,Halmos-von Neumann theorem, Radon measure,Dirac measure,Wendellocalization theory, theorem, measure.

10 , Banach algebra is indicated, the resolvent set, the resolvent function, theStone-Weierstrass theorem, the exchange of c *- algebraic characterization,Stone-Cech tight, and Gelfand-Naimark-Segal structure.

11 , Normal operator spectral theorem for continuous form of functional operation, operator of the absoluteFugledetheorems, spectral theorem of formal operators Borel functional forms, spectral projection computation,Weyl-von Neumann theorem, Banach The strong topology and the weak topology, algebra Banach Algebra of amplification, von Neumann Dual commutator theorem proof, \sigma- Qiang Tuo flutter, w*- Topology, \sigma- Weakly continuous functional operation.

12 , Von Neumann algebra in pre-dual, maximal Abelian algebra, severe free operator, operators of the spectral theorem for severe forms free operator, Atomic algebra and operator figure, close range, linear transformation operator, closed operator, heavy fixed operators, closed the resolvent operator sets, spectrum of unbounded operators.

13 , Unbounded symmetric operators, unbounded self-adjoint operators, essentially self-adjoint operators, basic criterion for self-adjoint operators, spectral theory of unbounded self-adjoint operators, projection-valued measure, strongly continuous one-parameter unitary groups,Stone theorem,von Neumann theorem, self-adjoint operators of Exchange, canonical commutation relations, Weyl relations.

Functional Analysis

1. Peter D. Lax, Functional Analysis, Wiley-Interscience, 2002. 　

1 , A.Ya.Helemskii , Lectures on functional analysis, continuous mathematics education centre in Moscow, 2004 。

2 , A.N.Kolmogorov 、 S.V.Fomin , Theory of functions and functional analysis, science press, 1989 。

3 , V.S.Vladimirov , Generalized functions in mathematical physics, science press, 1979 。

4 , A.A.Kirillov 、 A.D.Gvishiani , Theory and problems of functional analysis, science press, 1988 。

5 , Frigyes Riesz 、 Bela Szokefalvi–Nagy , Functional Analysis , Dover Publications 。

Kolmogorov, “Elements of the Theory of Functions and Functional Analysis”

L.V.Kantorovitch, G.P.Akilov “Functional Analysis”

А.Б. Antuonieweiqi functional analysis of problem sets higher education press

The functional analysis of theoretical questions and problems Kriloff

H.Brezis “Analyse Fonctionelle”

Conway, ” A Course in Functional Analysis “
Rudin, ” Functional Analysis “

Yoshida, “Functional Analysis”

Lang, “Real and Functional analysis”

I. M. Gelfand, “Generalized Functions” I-V

N.Bourbaki “Topological Vector Space”Chpt. 1-5

H.H.Schaefer, “Topological Vector Spaces”

J.L. Kelley, I. Namioka, “Linear Topological Spaces”

P.R. Halmos, “A Hilbert Space Problem Book”

I.E. Segal, R.A. Kunze, “Integrals and Operators”

Dunford,Schwarz, “Linear Operators”I

S.K. Berberian, “Lectures in Functional Analysis and Operator Theory”

J.M.Bony, Ergodic theory (ergodic theory) ,” That is the real measure theory “(J.M.Bony).

Zhang Gongqing, lectures on the functional analysis (upper and lower), Peking University
Xia Daoxing, of the real variable function theory and functional analysis (ⅱ), higher education
Xia Daoxing , Yan shaozong , Shu Wuchang , Tong Yusun ” A second course of functional analysis”

Liu peide of the functional analysis of the Wuhan University Press

Zheng Weihang an outline of the real analysis and functional analysis (ⅱ) the higher education press

Wang Lin ” Functional analysis of the counter example”

Xia Daoxing , YEUNG Ah-Li ” On topological linear spaces”  

165 The real analysis and functional analysis, Guo Dajun et al

“The problem sets and books”

165 The functional analysis of problem sets and solutions ( India ) V. K. Krishnan The

166 Of the theory of functions and functional analysis of the Cole Mo geluofu the

167 Of the functional analysis, problem analysis and solving method of the Qinghua Sun, Sun h a

168 The content, methods and techniques of functional analysis of Qinghua Sun , Hou Qian , Sun h a

The General liusitieernike of functional analysis, Sobolev

The functional analysis of problem set antuonieweiqi

The functional analysis of theoretical questions and problems Kriloff

“Increase”

169 Wang Lin of the counterexample to the functional analysis of the

170 Functional analysis of the new GUI of the speaking

173 The functional analysis : Theory and application :theorie et applications 》 Haim Brezis The

The application of theory of functions and functional analysis

1 Complex Hilbert On the space of self-adjoint operators and their effect on cyclic vector, spectrum of complex-valued function operators, square-integrability and the finite countably additive Borel Measure the relationship.

2 , And can be divided into Hilbert On the space of self-adjoint operators, as operators in the preceding paragraph at most countable direct sum.

3 Complex Hilbert Space Von Neumann Theorem.

4 Complex Hilbert On the space of self-adjoint operators and is the space of complex-valued functions defined on the real line bounded measurable functions on the product operator is unitarily equivalent.

5 , The spectral decomposition of bounded self-adjoint operator.

6 , Hilbert Space and complex Hilbert Of the space of bounded linear operators on a composite, closed operator, Hilbert Self-adjoint and full of self-adjoint operators on the space, linear quadratic composite proof of existence of closed operator.

7 , Criterion for self-adjointness of symmetric operators.

8 And construction of a locally convex space, projection and inductive limits, and direct sum, tensor products of locally convex spaces.

9 Topological linear spaces.

10 Vector space weak topology, the vector space of linear functionals on the space.

11 And deficiency indices of symmetric operators.

12 , Von Neumann Formula.

13 Symmetry symmetric expansion of operator, for example.

14 And the equivalence of the deficiency indices of symmetric operators Von Neumann Criterion.

15 , Spectrum of symmetric operators.

16 , Cayley Transform.

17 , Spectral theory of unbounded self-adjoint operators, Stone Theorem Tauber Theorem.

18 , Quadratic, the convergence of unbounded operators, Trotter Chernov Theorem.

19 , Self-adjoint operators perturbed, Friedrichs Expansion Kato Inequalities, Kalf-Walter-Schmincke-Simon Theorem Davies-Faris Theorem.

20 And commutator theorem.

21 , Semigroups, infinite generator function.

22 , Ergodic theory, ergodic theory and continuous flow.

23 Heat equation Schrodinger Equation Feynman Formula.

24 , Feynman-Kac Formula.

25 And the axioms of quantum mechanics.

26 , Bell Inequality and uncertainty of classical probability model of quantum mechanics.

27 Quantum information and quantum computation provides an overview.

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2 , M.Reed 、 B.Simon , Methods of Modern Mathematical Physics , Vol 2 , Academic Press , 1975 。

3 , N.Danford 、 J.T.Schwartz , Linear Operators , Interscience , 1963 。

4 , Е.В.Dаvies , One-Parameter Semigroups , Academic Press , 1980 。

5 , O.G.Smolyanov 、 E.T.Shavgulidze Land path integrals, Moscow University Press, 1990 。

6 , R.Alicki 、 M.Fannes , Quantum Dynamical Systems , Oxford University Press , 2001 。

7 , V.I.Smirnov Higher maths course, volume v, physical and mathematical books publishing house.

8 , V.I.Bogachev Gaussian measure, science press, 1997 。

9 , O.G.Smolyanov 、 A.Truman , Bell Inequality and probabilistic models of quantum systems, Russian Academy of science report 397 , 1 , 2002 ,