I've decided to learn real analysis for the first exam about this subject at university but I'm not sure about which book to buy.  Please, can you suggest a good reference book?
During the next quarter at uni, I'll be taking a course in real analysis and since I prefer studying with an additional text I thought I'd come here to look for some book recommendations.  My background: I'm comfortable with linear algebra and single variable calculus, but shakey on multivariable calculus. I did have a slight introduction in to geometry.  For linear algebra I found the common recommendation of Hoffman &amp; Kunze to be very helpful (although the abstractness came as a bit of a kicker at first).   For Single variable calculus I used the book Calculus by Adams and Essex, which I found frustrating to work with, because it often left me guessing at what they were trying to do or why they were doing it. I also felt that they exercised a certain lack of rigour. (Which may be due to the subject which would make it hard to derive from axioms.)  Details: Though it would be a complementary text, I'd prefer one which will still hold value as a work of reference at a later point in my studies. I've heard that Rudin's Principles of mathematical analysis is one of the better textst out there, but also read several comments discouraging Rudin's book as a first introduction to real analysis, hence my quest to gather more information and maybe get some more personalised recommendations.  The course: The mandatory course literature will consist of lecture notes, but 4 texts are suggested as recommended reading.   T.M. Apostol, Mathematical analysis. Addison-Wesley (1974) J. Dieudonné, Foundations of Modern Analysis. Academic Press (1960) A. van Rooij, Analyse voor Beginners.Epsilon Uitgaven, no. 6 (2003)   R.S. Strichartz, The way of analysis (1995)   Thanks in advance!
A problem about function N I will be attending a summer school which will last 10 days.  I want to know: for applying for a Schengen visa, is the purpose of my trip considered as touristic or study? How can I identify what application is using a given file? JSF convertDateTime renders the previous day Ok, so I'm such a newbie to Ubuntu and I bought a Windows 8 pre-installed laptop last month.  I updated to Windows 8.1 and then I thought about installing Ubuntu as a dual boot so I could mess around and learn more about it.   So I followed a Youtube tutorial (  ) and I got my stuff working fine.   The first few times I booted I got the GRUB menu instead of my default HP Boot OS Manager, and I was able to select my OS.   So I went to sleep and the next day I turned on my computer and the GRUB menu did not show up. I tried several times and it didn't automatically show up.  In order for me to see the GRUB menu I had to turn on my PC and on start had to press ESC to pause startup and press F9 to get boot options. Then from there I had to pick from OS Boot, Ubuntu, Ubuntu (Yes there were two Ubuntus available) and a default EFI file thingy.   When I click the first Ubuntu I get the GRUB Menu (I was too scared to try the second in case I screwed my laptop up) and I can safely load Ubuntu from there and use it (although I do have to increase my brightness every time I load Ubuntu because it somehow reduces my brightness to complete darkness on boot)  So my problem here is why isn't my GRUB showing on boot, after it worked on the first day?   My laptop is a HP TouchSmart j-078CA.  EDIT 1: FYI I just switched to Ubuntu LiveCD to check my Installation and found out that one of my partition is flagged as msftres. Here is a screenshot of my GParted on a LiveCD: When I try to create a reaction from chemmacros/reactions I got the following error. I dunno what I am missing. Please help. Thank you.     line 11: Undefined control sequence. \end   line 11: Missing number, treated as zero. \end   line 11: Missing number, treated as zero. \end     Minimum Working Example    \documentclass[12pt,a4paper]{article} \usepackage{chemmacros} \chemsetup{modules={reactions}} \begin{document} \begin{reaction}     A + B -&gt; C \end{reaction} \end{document} Restarting with new account on iOS/GameCenter The probability that an initial quantum state $|\psi_i\rangle$ becomes the final quantum state $|\psi_f\rangle$ is given by  \begin{eqnarray} P(i \rightarrow f) &amp;=&amp; |\langle\psi_f|\psi_i\rangle|^2 \tag{1}\\ &amp;=&amp; \langle\psi_f|\psi_i\rangle^*\langle\psi_f|\psi_i\rangle \\ &amp;=&amp; \langle\psi_i|\psi_f\rangle\langle\psi_f|\psi_i\rangle. \end{eqnarray}  Equation (1) seems to show that the probability for the transition ($i\rightarrow f$) can be interpreted as the system both moving forward in time ($i\rightarrow f$) with amplitude $\langle\psi_f|\psi_i\rangle$ and backward in time ($f\rightarrow i$) with amplitude $\langle\psi_i|\psi_f\rangle$ simultaneously.  Does this reasoning help to explain the Born rule? (Is it like the Transactional Interpretation of QM?)  I guess we must experience the macroscopic direction of time ($i\rightarrow f$) in accord with increasing entropy in an expanding universe whereas microscopically QM works both forwards and backwards in time.  Addition  This is an improved version of the argument including time-evolution operators.  The probability that an initial quantum state $|\psi_i\rangle$ evolves to become the final quantum state $|\psi_f\rangle$ is given by  \begin{eqnarray} P_{i \rightarrow f} &amp;=&amp; |\langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle|^2 \tag{2}\\ &amp;=&amp; \langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle^*\langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle \\ &amp;=&amp; \langle\psi_i|U^\dagger_{i \rightarrow f}|\psi_f\rangle\langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle \\ &amp;=&amp; \langle\psi_i|U_{f \rightarrow i}|\psi_f\rangle\langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle \end{eqnarray} where $U_{i \rightarrow f}$ is the forward-time evolution operator and $U_{f \rightarrow i}=U^\dagger_{i \rightarrow f}$ is the corresponding backward-time evolution operator.  Equation (2) seems to show that the probability $P_{i\rightarrow j}$ can be interpreted as the system first evolving forwards in time and then evolving backwards in time.  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