实分析

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• graycat · 2016 年

  ……有理数时为 1，否则为 0。那么 f 在 [0,1] 上的黎曼积分不存在，但测度论积分存在，结果为 0。至少有两本现在很经典的书，Royden 的《Real Analysis》和 Rudin 的《Real and Complex Analysis》。但还有更多，而且可能还能写出更多。对于 Rudin，可以忘了后半部分关于复变函数的内容。Royden 更容易读。Rudin……

  …rational and 0 otherwise. Then the Riemann integral of f over [0,1] does not exist, but the measure theory integral does and gives 0 for the result. There are at least two now classic books, Royden, Real Analysis and Rudin, Real and Complex Analysis. But there are more, and likely more can be written. For Rudin, can f'get about the last half on functions of a complex variable. Royden is easier to read. Rudin…

• graycat · 2019 年

  建议：先把复变函数放一边，改学测度论。傅里叶变换和拉普拉斯变换，就用测度论来讲。测度论方面，用 H. Royden 的《Real Analysis》，加上 W. Rudin 的《Real and Complex Analysis》前半本。然后上一门基于测度论的研究生概率论课。好作者有 Loeve、Neveu、Breiman、Chung 等等。……

  Suggestion: Set aside complex variables and do measure theory instead. For the Fourier and Laplace transforms, cover those via measure theory. For measure theory, H. Royden, Real Analysis and the first half of W. Rudin, Real and Complex Analysis. Then take a course in graduate probability which is based on measure theory. Good authors are Loeve, Neveu, Breiman, Chung, among others.…

• graycat · 2014 年

  ……做完了；再学一些在后续数学分析里很重要的内容。Spivak 的《Calculus on Manifolds》。上面这三本一度是哈佛著名的 Math 55 的主要参考书。Royden 的《Real Analysis》，讲测度论，并开始涉及泛函分析。优雅。Rudin 的《Real and Complex Analysis》，非常扎实，又是测度论，泛函分析讲得更多。还有 von Neumann 那个关于……的巧妙证明

  …done; learn some more material important in the rest of mathematical analysis. Spivak, Calculus on Manifolds The three above were at one time the main references for Harvard's famous Math 55. Royden, Real Analysis Measure theory and a start on functional analysis. Elegant. Rudin, Real and Complex Analysis Rock solid, measure theory again, and more on functional analysis. Also von Neumann's cute proof of the…

• graycat · 2015 年

  ……这本书现在有英文版了）。可能你对相对论或者数学物理的其他部分不感兴趣——没关系，这反而能帮你把一些主题排除掉。然后，像上面说的，Royden 的《Real Analysis》和 Rudin 的《Real and Complex Analysis》前半本，再加上其他几个可选的替代方案，覆盖测度论和泛函分析的入门。测度论把微积分重做了一遍，而且更有力……

  …book is available in English now). It may be that you are not interested in relativity theory or the rest of mathematical physics -- fine, and that can help you set aside some topics. Then, Royden, Real Analysis and the first half of Rudin's R&CA as above, along with any of several alternatives, cover measure theory and the beginnings of functional analysis. Measure theory does calculus again and in a more powerful…

• graycat · 2015 年

  ……测度论是学 Neveu 的好前置。Rudin 的《Principles》稍微少了点。取而代之，Rudin 的《Real and Complex Analysis》前半本，也就是实分析那一半，是很好的前置。Royden 的《Real Analysis》也是。Neveu 优雅得难以置信，但 Breiman 的《Probability》，那本 SIAM 出的书，有平装本，相当不错，通常比 Neveu 容易，没那么优雅，更贴近应用，而且没有一些……

  …Measure Theory is a good prerequisite to Neveu. Rudin, Principles is a bit too little. Instead, the first half, the real half of Rudin's Real and Complex Analysis is a good prerequisite. So, is Royden's Real Analysis. Neveu is elegant beyond belief, but Breiman, Probability, the SIAM book, available in paperback, is darned good, usually easier than Neveu, less elegant, closer to applications, and without some of…

• graycat · 2016 年

  ……用别的函数逼近函数，比如，怎么用别的随机变量逼近随机变量，比如，学傅里叶理论），但为此也得有测度论，比如从 Royden 的《Real Analysis》和/或 Rudin 的《Real and Complex Analysis》前半本里学。为此，就像 Math 55 那样，前置是 Rudin 的《Principles》。Rudin 的《Functional Analysis》还要更进阶。我被研究生院录取了……

  …approximate functions by other functions, e.g., how to approximate random variables by other random variables, e.g., learn Fourier theory), but for that also should have measure theory, e.g., from Royden, Real Analysis and/or the first half of Rudin, Real and Complex Analysis. For that, as in Math 55, a prerequisite is Rudin's, Principles. Rudin's Functional Analysis is more advanced, still. I got accepted to grad school…

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