At this stage you should have learnt limits, continuity and convergence, over the reals. These are central concepts of calculus in both one and several variables. These can be generalised. You will study the idea of several spaces (metric space, topological space, Banach, Hilbert), compactness, connectedness, linear operators, elementary Lebesgue theory and measure theory, etc.
Contents of analysis courses varies almost everywhere (a.e.). As you may see, the term analysis appear in the first and second year calculus to forth year analysis. Kolmogorov and Fomin's text fits this stage well, they cover most of the above topics. Apostol, Marsden and Hoffman, and Simmons make good references. Several books entitled analysis are indeed advanced calculus (stage two calculus) or not aimed for mathematicians. Stay away from Mathematical analysis for business and finance or the like.
Only for metric spaces and topological spaces (also consult the list of Topology below):
- Burkill J.C. and Burkill H. A Second Course in Mathematical Analysis
- Giles J.R. Introduction to the Analysis of Metric Spaces - One of the Australian Mathematical Society Lecture Series, gives nice discussion on limit processes, continuity, compactness and metric topology.
- Searcóid M.Ó. Metric Spaces
- Sutherland W.A. Introduction to Metric and Topological Spaces
Also deals with Lebesgue theory, Banach spaces and Hilbert spaces, etc:
- Apostol T.M. Mathematical Analysis - If you enjoy his Calculus Vol. 1 in stage 1 and Calculus Vol. 2 in stage 2, then you should go for this. Solution by Shin-Yi Lee can be found here.
- Bass R.F. Real Analysis (Measure Theory) [.pdf] (FREE!)
- Beals R. Analysis: An Introduction - It covers a lot of stuff and hence too concise for an introductory text, maybe. Treat this as a reference.
- Bruckner A.M., Bruckner J.B.and Thomson B.S. Real Analysis (here for FREE!)
- Chen W.W.L. Introduction to Lebesgue Integration (FREE!)
- Chen W.W.L. Linear Functional Analysis (FREE!)
- Cohen G.L. A Course in Modern Analysis and its Applications
- Davidson K.R. and Donsig A.P. Real Analysis with Real Applications
- Jost J. Postmodern Analysis
- Kolmogorov A.N. and Fomin S.V. Elements of the Theory of Functions and Functional Analysis - The material is based on lectures given by the authors in the Moscow State University.
Kolmogorov A.N. and Fomin S.V. Introductory Real Analysis - This version is translated and edited by R.A. Silverman. It seems that this is a more popular edition. See the typo by Edgar [.pdf].
- Marsden J.E. and Hoffman M.J. Elementary Classical Analysis - This book sits somewhere between the stage 2 calculus and stage 3 analysis in this list. One feature is that proofs are often left behind examples.
- Rudin W. Principles of Mathematical Analysis - Baby Rudin. A modern classic. Get a copy and play around with it. See also George M. Bergman's supplement exercises [.ps], notes [.ps] and q&a; and Evelyn Silvia's Companion Notes.
- Santos D. Real Analysis/Advanced Calculus [.pdf] (FREE!)
- Saxe K. Beginning Functional Analysis - Large proportion of this book can be studied at this stage.
- Simmons F.G. Introduction to Topology and Modern Analysis - Nice mathematical writing.
- Sternberg S. Advanced Calculus [.pdf] (FREE!)
- Tao T. Analysis 2 (FREE!)
- Wilkins D.R. General Topology and Real Analysis (FREE!)