Re: analysis suggetion
I suggest a recent book on analysis:
Mathematical Analysis: A Concise Introduction. The author has taken
great care in providing many aids in the first four chapters and only then began to remove the scaffolding.
While Rudin is a classic, let us not forget the mention in its preface that "it is meant for first year graduate or last year undergraduate students." Furthermore, most people who worked through this book did so with the help of a teacher, making it all more reasonable. Tackling Rudin by yourself is a pretty difficult, to say the least, enterprise.
Back to Schröder, it is a comprehensive book. You'll get exposure to all of analysis of one variable, including numerical methods (giving you a taste of numerical analysis), and then moving on to more general settings. The author has explicitly stated that the emphasis is on the
methods of real analysis, particularly those that generalize to other contexts. Therefore, most of what you learn is applicable directly
mutatis mutandis. I
f you are lacking in motivation, part three of the book is named Applied Analysis, furnishing many examples in diverse areas. It starts off with physics, going through harmonic oscillators, Maxwell's equations, heat equation and diffusion PDEs to name a few, passes by ordinary differential equations in Banach spaces and ends with the Finite Elements Method. He advises the interested readers to go straight to those chapters to have an idea of what can be done.
Amidst part one you have small bits of the theory of Lebesgue integration intervened with the Riemann-Stieltjes integral, showing you what are each strengths and providing insight of why certain definitions and theorems will appear in part two. Part two is where the generalization begins and you get to reap benefits from your efforts in analysis in one variable: he discusses vector spaces, metric spaces, normed spaces and inner product spaces. He gives a thorough explanation of metric spaces topology, which makes for a long but useful chapter. He then proceeds to construct measure spaces and integration in more abstract settings, but since you had such good guidance in one variable and given his focus on methods, many proofs are labelled "see theorem X.Y", and more often than not you will see it is almost copy and paste. In this part you will get a taste of Measure Theory, a slight Introduction to Differential Geometry and Hilbert Spaces, thus demonstrating how analysis isn't an island but a coherent continent connected with many areas of mathematics. (Happy)
