Stage 3 Introductory Analysis

Introductory Analysis:
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):

Also deals with Lebesgue theory, Banach spaces and Hilbert spaces, etc: