Which are the Best Books for Mathematical Analysis After Rudin and Royden?

After mastering Rudin and Royden, finding the right next step in mathematical analysis can be a daunting task. Different books offer a range of depth and focus, suitable for learners at various stages of comprehension and interest. This article explores some highly-regarded texts that can further your understanding of advanced mathematical analysis.

Real Analysis by Terry Tao

If you've already gone through Rudin and found it particularly dry, and are struggling to get a holistic sense of what’s happening, Real Analysis by Terence Tao might be the perfect choice. Tao, a renowned mathematician, takes a unique approach to the subject, focusing on sequences rather than the traditional epsilon-delta characterizations of continuity, uniform continuity, and other concepts.

Tao's approach emphasizes natural methods to tackle problems, making complex ideas more accessible. The book provides a smooth transition for those who are getting introduced to analysis, but it's also incredibly rewarding for those looking to deepen their understanding after a foundational course. He dives into topics such as metric topology of subsets of normed linear spaces, including sequence spaces and function spaces, offering a natural and intuitive understanding.

Additionally, Tao explores the correspondence between the topology of a metric space (M) and the space of continuous functions on (M), denoted as (C(M)). He helps readers understand one space given the other, providing a rich and engaging exploration of these concepts. The book’s approach to measure theory is particularly noteworthy: Tao doesn't delve into abstract measure theory but focuses on the Lebesgue measure on (mathbb{R}), providing deep insights and detailed explanations.

What sets Tao's book apart is the quality of his writing, clarity in explanations, and the logical progression of ideas. His notes from his blog “An epsilon of room” are also highly valuable, offering further insights and examples that complement the main text.

Additional Resources

Tao's book is part of a larger suite of resources, including his blog and notes, which are incredibly helpful for anyone serious about mathematical analysis. These resources provide additional perspectives and deeper dives into the topics discussed in the book.

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Conclusion

Choosing the right book for mathematical analysis after Rudin and Royden depends on your specific interests and background. For those who prefer a natural and intuitive approach, Tao's Real Analysis is an excellent choice. It offers both a solid foundation and advanced insights, making it a valuable resource for both beginners and advanced learners. Explore these texts and notes to further your understanding of advanced mathematical analysis.