Garling D.J.H. A Course in Mathematical Analysis. Vol. 2: Metric and Topological Spaces, Functions of a Vector Variable

Cambridge University Press, 2014. — 336 p. — ISBN: 9781107675322.
This second volume goes on to consider metric and topological spaces. Topics such as completeness, compactness and connectedness are developed, with emphasis on their applications to analysis. This leads to the theory of functions of several variables: differentiation is developed in a coordinate free way, while integration (the Riemann integral) is established for functions defined on subsets of Euclidean space. Differential manifolds in Euclidean space are introduced in a final chapter, which includes an account of Lagrange multipliers and a detailed proof of the divergence theorem.Part I. Metric and Topological Spaces:
Metric spaces and normed spaces.
Convergence, continuity and topology.
Topological spaces.
Completeness.
Compactness.
Connectedness.
Part II. Functions of a Vector Variable:
Differentiating functions of a vector variable.
Integrating functions of several variables.
Differential manifolds in Euclidean space.
Appendix A. Linear algebra.
Appendix B. Quaternions.
Appendix C. Tychonoff's theorem.