泛函分析
Functional Analysis
Walter Rudin · 初版 1973 年
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本书为本科课程而写，这一新版涵盖了当前的研究课题，并包含更多习题和示例。新增课题包括：角谷不动点定理、洛蒙诺索夫不变子空间定理，以及一个遍历定理
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Functional Analysis
Walter Rudin · 初版 1973 年
本书为本科课程而写，这一新版涵盖了当前的研究课题，并包含更多习题和示例。新增课题包括：角谷不动点定理、洛蒙诺索夫不变子空间定理，以及一个遍历定理
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……定理，但希尔伯特空间中的同样事情更容易。在某种意义上，L1 和 L-无穷互为对偶，但 L2 是自对偶的，这更好。填补所有这些细节及更多内容是泛函分析 101 的一部分。很难错过 W. Rudin 的至少三本书：《Principles of Mathematical Analysis》、《Real and Complex Analysis》和《Functional Analysis》。还有更多，但我得先解决一些 bug……
…theorem, but the same thing in Hilbert space is easier. There is a sense in which L1 and L-infinity are duals of each other, but L2 is self-dual which is nicer. Filling in all these details and more is part of functional analysis 101. There tough to miss at least three books of W. Rudin: Principles of Mathematical Analysis, Real and Complex Analysis, and Functional Analysis. There's more, but I've got some bugs to get out of…
……H. Olmsted，《分析中的反例》 John C. Oxtoby，《测度与范畴：拓扑空间与测度空间之间类比的研究》 Walter Rudin，《实分析与复分析》 Walter Rudin，《泛函分析》 Leo Breiman，《概率论》 Kai Lai Chung，《概率论教程》第二版 Jacques Neveu，《概率演算的数学基础》 Erhan Cinlar，《随机过程导论》……
…H. Olmsted, Counterexamples in Analysis John C. Oxtoby, Measure and Category: A Survey of the Analogies between Topological and Measure Spaces Walter Rudin, Real and Complex Analysis Walter Rudin, Functional Analysis Leo Breiman, Probability Kai Lai Chung, A Course in Probability Theory, Second Edition Jacques Neveu, Mathematical Foundations of the Calculus of Probability Erhan Cinlar, Introduction to Stochastic Processes…
……Radon-Nikodym 定理（微积分基本定理的成熟版本，具有惊人的威力）和傅里叶积分。如果你不读后半部分，我不会扣分。Rudin 的《Functional Analysis》——可以了解分布。
…Radon-Nikodym theorem (grown up version of the fundamental theorem of calculus and with astounding power) and the Fourier integral. I won't take off points if you don't read the second half. Rudin's Functional Analysis -- get to see distributions.
……关于自伴矩阵基本结果的逆函数定理和隐函数定理，在 Rudin 的《数学分析原理》（Principles of Mathematical Analysis）第三版里作为不错的习题给出……而谱理论在 Rudin 的《泛函分析》（Functional Analysis）里。还能额外收获对广义函数的漂亮处理，也就是替代量子力学里狄拉克 δ 函数的用法。至于如何得到自伴……的特征值和正交特征向量的结果……
…on the basic results of self-adjoint matrices are the inverse and implicit function theorems given as nice exercises in the third edition of Rudin's Principles .... And spectral theory is in Rudin's Functional Analysis. Also get a bonus of a nice treatment of distributions, that is, replace the Dirac delta function usage in quantum mechanics. For how to get the eigen value and orthogonal eigen vector results for self-adjoint…
……另一本书，很薄，Halmos 写的，《Introduction to Hilbert Space and the Theory of Spectral Multiplicity: Second Edition》（Dover Books on Mathematics），大概 10 美元。要更多的话，有 W. Rudin 的《泛函分析》（Functional Analysis），一如既往地精确得惊人。在任何大学里都很难找到一位物理教授能把傅里叶变换或谱理论讲得那么好。所以，可以……
…in another book, short, by Halmos, in Introduction to Hilbert Space and the Theory of Spectral Multiplicity: Second Edition (Dover Books on Mathematics) for about $10. For more, there is W. Rudin, Functional Analysis, as usual for Rudin, with astounding precision. Will have a tough time at any university finding a physics prof who will do that well with either the Fourier transform or spectral theory. So, can get…
……定理，并在更一般的情形下定义。正如 A. Kolmogorov 1933 年的一篇论文，勒贝格积分给概率论、统计学和随机过程提供了坚实的基础。在 W. Rudin 的《泛函分析》（Functional Analysis）里能得到对广义函数的处理，把物理学想用狄拉克 δ 函数做的事收拾干净，还涵盖谱理论。从那儿就可以奔博士学位去了。为此主要需要 (A) 通过……
…theorems and is defined in more general situations. As in a 1933 paper of A. Kolmogorov, the Lebesgue integral gives a solid foundation to probability, statistics, and stochastic processes. In W. Rudin, Functional Analysis get a treatment of distributions that cleans up what physics tries to do with the Dirac delta function and also covers spectral theory. From there, can go for a Ph.D. For that will need mostly (A) pass…