Trinity College Dublin

MA2321 - Analysis in Several Real Variables
Dr. David R. Wilkins
Course Material from the Academic Year 2015/16

Notes for Sections 1, 2 and 3 of module MA2321 in Michaelmas Term 2015: These notes cover ordered fields, the real number system, limits and covergence of sequences of real numbers, the Bolzano-Weierstrass Theorem, limits and continuity for functions of a real variable, the Extreme Value Theorem and uniform continuity for functions of one variable.
Notes for Sections 4 and 5 of module MA2321 in Michaelmas Term 2015: These notes cover differentiability and smoothness for functions of a single real variable, and the theory of the Riemann integral (or Darboux integral) in one dimension.
Notes for Section 6 of module MA2321 in Michaelmas Term 2015: These notes cover convergence and limits for sequences of points in Euclidean spaces, and limits and convergence for functions of several real variables, and also include basic results concerning open and closed sets in Euclidean spaces.
Notes for Section 7 of module MA2321 in Michaelmas Term 2015: These notes cover the theory of differentiability for functions of several real variables.

Additional sections developing the material further are available here.

Notes for Section 8 of module MA2321 in Michaelmas Term 2015: These notes begin by discussing continuous differentiability and smoothness for functions of several real variables, preparing for the statement and proof of the Inverse Function Theorem. Various corollaries of the Inverse Function Theorem are then discussed, including in particular the Implicit Function Theorem. The notes conclude by proving the equivalence of two criteria that characterize smooth submanifolds of Euclidean spaces.
Notes for Section 9 of module MA2321 in Michaelmas Term 2015: These notes include a brief discussion of topological spaces in general, linking in with the results concerning open and closed sets in Euclidean spaces that were proved in Section 4. The definition and basic properties of compact topological spaces are discussed. The multidimensional Bolzano-Weierstrass Theorem is used to prove the existence of Lebesgue numbers for open covers of closed bounded subsets of Euclidean spaces, and this result in turn is used to prove the multidimensional Heine-Borel Theorem, which states that a subset of a Euclidean space is compact if and only if it is both closed and bounded. This is followed by a discussion of various results in the theory of metric spaces that relate to compactness. The notes conclude with a discussion of normed vector spaces, including a proof that all norms on a finite-dimensional real vector space are equivalent.