Just finished Spivak. How does real analysis differ? I just finished working through and understanding everything in Calculus by Michael Spivak. Now I'm contemplating reading Rudin, since it's said commonly that real analysis is the next step. I just realized though, I don't really understand what Rudin has that Spivak doesn't. Calculus formally defines limits, convergence, derivatives, integrals and everything. How exactly does Rudin (and real analysis in general) differ? I previously thought they differed roughly by rigor, but Spivak has covered what I previously thought was real analysis.
 A: There are plenty of concepts from real analysis that aren't in Spivak's text. There are two main concrete directions to go, and a variety of abstract directions to go.
First there are function spaces. The main important concept that function spaces introduce is that there are different kinds of convergence which have different properties.
One of the most important examples is as follows. Suppose $f_n$ is a sequence of continuous functions on $[a,b]$. If they converge uniformly to $f$, i.e. $\lim_{n \to \infty} \sup_{x \in [a,b]} |f_n(x)-f(x)| = 0$, then $f$ is continuous. But if they converge pointwise, i.e. for every $x$ we have $\lim_{n \to \infty} f_n(x) = f(x)$, then $f$ may not be continuous. The classic counterexample is $f_n(x) = x^n$ on $[0,1]$.
Second there is Lebesgue integration. It turns out that the Riemann integration studied in calculus has bad analytic properties. A nice property that we very often want is that if $f_n \to f$ pointwise, then $\int f_n dx \to \int f dx$. Riemann integration makes it very difficult to guarantee this property. In particular it is very difficult to guarantee that the right hand side even makes sense. Lebesgue integration defines integration in such a way that the right hand side will always make sense, and gives a number of convenient criteria to guarantee that the convergence I described above will occur. As a bonus, a huge wealth of new functions become integrable, and a framework for integration over spaces other than $\mathbb{R}^n$ appears.
The abstract directions are numerous. However, they usually begin with metric spaces. I will let someone else discuss the relevance of this topic.
A: It sounds like you're doing this for self-study. Personally, I would recommend against baby Rudin (Principles of Mathematical Analysis) and even moreso green Rudin (Real & Complex analysis). It's not that Rudin isn't good -- he's great -- but I don't think it's a good progression. I'll suggest a few directions for you to go in. 


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*Spivak's Calculus on Manifolds. This will require you know some basic linear algebra, and you might have to check a few definitions, but this might be an exciting continuation of your course on calculus. You would be familiar with the style of writing, and more importantly, the book is absolutely fantastic. Multivariable calculus really takes on a life of its own, and you can really feel the full force of everything you've just learned with immense applications to real-life phenomena, especially electromagnetism. To this end, Div, grad, curl and all that is an equally charming book that could whet your appetite. 

*Stein and Shakarchi's Princeton lectures in analysis, Vol. 1 : Fourier Analysis. The text is very readable, and most importantly, doesn't require you to know about Lebesgue integration. Moreover, Fourier analysis motivates a lot of analysis. You might get lost pretty quickly, but then you get start going through an introductory analysis textbook (like Bartle and Sherbert). 

*Something completely different! Analysis is just one part of mathematics. I think it's a great idea to get an early exposure to group theory. I learned from John & Margaret Maxfield's Abstract algebra and solutions by radicals. It introduces you to groups, permutations, Cayley's theorem, the basics of rings and fields and even field extensions, all in a pretty accessible short book. 

