
MA2321  Analysis in Several Real Variables
Dr. David R. Wilkins
Lecture Slides  Michaelmas Term 2017

Lecture Material
 Lecture 1 (September 25, 2017)
 This lecture covered the definition of an
ordered field, the Least Upper Bound Principle,
and the characterization of the
real number system as a Dedekindcomplete ordered
field.
 Lecture 2 (September 28, 2017)
 This lecture covered basic results concerning
convergence of infinite sequences of real numbers,
including the onedimensional case of the
BolzanoWeierstrass Theorem.
 Lecture 3 (September 28, 2017)
 This lecture covered basic results concerning
the scalar product and Euclidean norm of vectors
in ℝ^{n}, including
Schwarz's Inequality and the Triangle Inequality.
The definition of convergence was presented for
sequences of points in Euclidean spaces
ℝ^{n} of arbitrary (finite)
dimension n. The lecture also presented
the definitions of limit points and
isolated points of subsets of
ℝ^{n}.
 Lecture 4 (October 2, 2017)
 This lecture covered versions of the
BolzanoWeierstrass Theorem and Cauchy's Criterion
for Convergence applicable to bounded sequences
of points in Euclidean spaces ℝ^{n}
of arbitrary (finite) dimension n.
 Lecture 5 (October 5, 2017)
 This lecture introduced the definition of open sets
in Euclidean spaces, and in subsets of Euclidean spaces,
and developed basic properties of the collection of
open sets in subsets of Euclidean spaces.
 Lecture 6 (October 5, 2017)
 This lecture presented a criterion for convergence
for a sequence of points in a Euclidean spaces, expressed
in terms of the open sets of that space, and not explicitly
referencing the distance function on that space. The
lecture continued with a discussion of closed sets in
subsets of Euclidean spaces.
 Lecture 7 (October 9, 2017)
 This lecture introduced the basic definitions of
limits and continuity for functions
of several real variables, and developed some basic
consequences of those definitions.
 Lecture 8 (October 12, 2017)
 This lecture developed further the theory of
limits of functions of several real variables.
 Lecture 9 (October 12, 2017) (not final)
 This lecture developed further the theory of
continuous functions of several real variables.
The lecture also presented applications of the
multidimensional BolzanoWeierstrass Theorem to prove
various results concerning continuous functions on closed
bounded subsets of Euclidean spaces, including a
multidimensional version of the Extreme Value Theorem,
the result that continuous functions defined over
such closed bounded sets are uniformly continuous,
and a proof of the equivalence of norms on a
finitedimensional vector space.
 Lecture 10 (October 19, 2017) (not final)
 This lecture presented the definition of the Riemann
integral for functions of a single real variable,
adopting the approach using Darboux upper and
lower sums.
 Lecture 11 (October 19, 2017) (not final)
 This lecture developed the theory of the Riemann
integral for functions of a single real variable.
 Lecture 12 (October 23, 2017) (not final)
ℝ^{n}
 This lecture developed the theory of the Riemann
integral for functions of a single real variable.
 Lecture 13 (October 26, 2017) (not final)
 This lecture discussed partitions of closed
ndimensional cells in
 Lecture 14 (October 26, 2017) (not final)
 This lecture presented the definition of the Riemann
integral for functions of several real variables.
 Lecture 15 (November 2, 2017) (not final)
 This lecture developed the theory of the Riemann
integral for functions of several real variables.
 Lecture 16 (November 2, 2017) (not final)
 This lecture developed the theory of the Riemann
integral for functions of several real variables.
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Dr. David R. Wilkins,
School of Mathematics,
Trinity College Dublin.