Intereting Posts

Probability for “drawing balls from urn”
Find unit vector given Roll, Pitch and Yaw
An short exact sequence of $\mathfrak{g}$ of which head and tail are in category $\mathcal{O}$.
Why “a function continuous at only one point” is not an oxymoron?
the number of Young tableaux in general
Monotone Class Theorem and another similar theorem.
An elementary question regarding a multiplicative character over finite fields
Homeomorphism between spaces equipped with cofinite topologies
Proving that $\lim_{x\to1^-}\left(\sqrt{1-x}\cdot\sum_{n=0}^\infty~x^{n^a}\right)=\Gamma\left(1+\frac1a\right)$
Proving that every isometry of $\mathbb{R}^n$ is of the form of a composition of at most $n+1$ reflections
Visualizing quotient groups: $\mathbb{R/Q}$
Fundamental Theorem of Calculus
Canonical basis of an ideal of a quadratic order
Convexity of a trace of matrices with respect to diagonal elements
Showing a Ring of endomorphisms is isomorphic to a Ring

I am interested in self-studying real analysis and I was wondering which textbook I should pick up.

I have knowledge of all high school mathematics, I have read How to Prove It by Daniel J. Velleman (I did most of the excercises) and I have completed a computational calculus course which covered everything up to and including integration by parts (including the substitution method and Riemann sums)

I am currently considering:

- Artinian - Noetherian rings and modules suggest study guide
- I want to get good at math, any good book suggestions?
- Beginning of Romance
- A good reference to begin analytic number theory
- Logic and number theory books
- Exercise books in functional analysis

- Principles of Mathematical Analysis by Walter Rudin
- Calculus by Michael Spivak
- Understanding Analysis by Stephen Abbott
- Mathematical Analysis by Tom M. Apostol

From what I have heard **Principles by Rudin** is not very well suited for self-study and that while the exercises are extremely difficult, if you take the time they are worth the effort.

I have heard that while **Calculus by Spivak** explains proofs in much more detail than **Principles**, it doesn’t cover all of the material in the latter.

I don’t know much about **Understanding Analysis by Abbott**, I have only seen some comments saying that it is an excellent introduction to analysis.

Extra clarification edit:

I would prefer a book that would not ”dumb down” the material, something that would not hold my hand through every step, something that would force me to fill in the gaps myself instead of explaining every single step. That is why I am currently leaning towards Rudin, but before I make the decision I would still like some information on the book by Apostol or any other options that might be suitable for me.

- Real analysis supremum proof
- Evaluation of $\int_{0}^{\infty} \cos(x)/(x^2+1)$ using complex analysis.
- Two definitions of Taylor polynomials
- Evaluation of $\int_0^1 \frac{\log^2(1+x)}{x} \ dx$
- Second Countability of Euclidean Spaces
- Prove that a bounded sequence has two convergent subsequences.
- An open interval as a union of closed intervals
- Why are norms continuous?
- How do I prove that $f(x)f(y)=f(x+y)$ implies that $f(x)=e^{cx}$, assuming f is continuous and not zero?
- An algebra (of sets) is a sigma algebra iff it is a monotone class

Rudin’s book is too abstract in some sense because it requires some knowledge or sense of metric topology. Although Rudin explains the basic theory, but I don’t think this is not appropriate to beginner.

Spivak’s calculus is “calculus”. Although it is quite tough, it is not a book for undergraduate analysis.

I recommend three books:

- Ross, Elementary Analysis: The Theory of Calculus
- Marsden and Hoffman, Elementary Classical Analysis
- Apostol, Mathematical Analysis

Ross helps reader to understand one dimensional real analysis. It gives quite good examples and appropriate exercise problem. But the book doesn’t cover multivariable things.

Marsden and Hoffman gives a tons of examples and interesting exercise problems. Although it is quite a challenge to reader, the book gives many pictures and good explanation to the subject. Although some part is based on higher dimensional setting, it is quite readable. I strongly recommend this book. If you read this book, you have to aware that the definition of compactness in this book is `sequential compactness’.

Finally, the book of Apostol gives almost full details to the proof. It covers many topics. Maybe this book is more appropriate to person who want to know more advanced topics.

I heard that one of my friends says Pugh’s Real Mathematical Analysis is good, but I didn’t read that book.

- Prove $\exists T\in\mathfrak{L}(V,W)$ s.t. $\text{null}(T)=U$ iff $\text{dim}(U)\geq\text{dim}(V)-\text{dim}(W)$.
- Why $\sqrt {-1}\cdot \sqrt{-1}=-1$ rather than $\sqrt {-1}\cdot \sqrt{-1}=1$. Pre-definition reason!
- Additivity of the matrix exponential of infinite matrices
- Asymptotic for primitive sums of two squares
- how prove $\sum_{n=1}^\infty\frac{a_n}{b_n+a_n} $is convergent?
- Do these $\sigma$-algebras on second countable spaces coincide?
- Arrangement of the word 'Success'
- Why is the ring of matrices over a field simple?
- linear transformation $T$ such that $TS = ST$
- Closed subsets of compact sets are compact
- Conditions Equivalent to Injectivity
- L : $\mathbb{R}^n \rightarrow \mathbb{R}^m$ is a linear mapping, linear independence of $L$ mapped onto a set of vectors.
- By substituting $z = h(t)$ show that $\delta(h(t))=\sum\limits_{i}\frac{\delta(t−t_i)}{\mid h^{\prime}(t_i)\mid}$
- Prove that $\sum\limits_{k=1}^{n-1}\tan^{2}\frac{k \pi}{2n} = \frac{(n-1)(2n-1)}{3}$
- Prove partial derivatives exist, but not all directional derivatives exists.