Module Code - Title:
MB4005
-
ANALYSIS
Year Last Offered:
2025/6
Hours Per Week:
Grading Type:
N
Prerequisite Modules:
MS4021
MS4022
Rationale and Purpose of the Module:
To develop an understanding of formal methods of mathematical analysis, as applied to sets, real numbers, and general topology.
Syllabus:
• Set theory: equivalence classes of sets, cardinal numbers, countability and uncountability, including the uncountability of R.
• Functions of a real variable: limits, continuity and differentiability from first principles.
• Multivariate functions: inverse function theorem, implicit function theorem.
• Complex functions: differentiability and Cauchy-Riemann equations.
• The completeness property: Bolzano-Weierstrass theorem, Cauchy sequences and completeness.
• Sequences and series of functions: pointwise and uniform convergence, term-by-term differentiation and integration.
• General topology: Euclidean n-space, metric spaces, connectedness, compactness, fixed point theorem, Hilbert spaces.
Learning Outcomes:
Cognitive (Knowledge, Understanding, Application, Analysis, Evaluation, Synthesis)
On successful completion of this module, students should be able to:
1. Test, and formally prove, the continuity and differentiability of functions.
2. Identify set equivalences and countable sets. Prove the countability of the rationals and the uncountability of the reals.
3. Prove and apply the theorems studied, including inverse function theorem, implicit function theorem, Bolzano-Weierstrass theorem and fixed point theorem.
4. Test a sequence or series of functions for pointwise or uniform convergence, and recognise implications for term-by-term differentiation and integration.
5. Define a metric space and use this concept to extend ideas in Euclidean spaces to abstract spaces.
Affective (Attitudes and Values)
None
Psychomotor (Physical Skills)
None
How the Module will be Taught and what will be the Learning Experiences of the Students:
Normal lecture and tutorial mode of delivery. Mid-term exam(s) and final exam.
Graduates should have a good grounding in the rigour and abstraction in
mathematics and have and an appreciation of aspects of mathematical analysis at a high level. Graduates should be able to engage in discussion with teachers and
mathematicians concerning the rigorous treatment of calculus and related topics.
Research Findings Incorporated in to the Syllabus (If Relevant):
Prime Texts:
Strichartz, R. (2000)
The Way of Analysis
, Jones and Bartlett
Apostol T. M. (1974)
Mathematical Analysis 2nd ed.
, Pearson
Ponnusamy, S. (2012)
Foundations of Mathematical Analysis
, Birkhauser
Other Relevant Texts:
Lipschutz, S (2011)
Schaum's Outline of
General Topology
, Schaum's Outlines
Mattuck, A.P. (2013)
Introduction to Analysis
, CreateSpace Independent Publishing Platform
Rosenlicht, M (2013)
Introduction to Analysis
, Important Books
Adams, R. A. (2013)
Calculus: A Complete
Course, 8th ed.,
, Prentice Hall
Stewart, J. (2012)
Calculus, 7th ed.
, Cengage Learning
Programme(s) in which this Module is Offered:
BSMSCIUFA - MATHEMATICAL SCIENCES
BAJOHOUFA - JOINT HONOURS
BSPHEDUFA - PHYSICAL EDUCATION
Semester(s) Module is Offered:
Autumn
Module Leader:
Clifford.Nolan@ul.ie