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

Distributed Printed Notes in Michaelmas Term 2015
 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 BolzanoWeierstrass 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 Material not covered in Lectures in Michaelmas Term 2015
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 BolzanoWeierstrass
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
HeineBorel 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 finitedimensional
real vector space are equivalent.
Full Notes in Michaelmas Term 2015
Assignments for Michaelmas Term 2015
Correspondences with MA2223 in Michaelmas Term 2015
There were a number of correspondence between the content of modules MA2223 and MA2321 summarized here.
Material to aid Preparation for the Annual Examination 2016
Back to D.R. Wilkins: Lecture Notes
Dr. David R. Wilkins,
School of Mathematics,
Trinity College Dublin.