Analysis in Euclidean and Metric Spaces

Singapore University of Social Sciences

Analysis in Euclidean and Metric Spaces (MTH402)

Synopsis

The techniques developed in MTH401 are exploited in spaces of continuous functions and applications are made to differential and integral equations. Differentiable aspects of vector-valued functions are then discussed.

Level: 4
Credit Units: 5
Presentation Pattern: Every July

Topics

The space of continuous functions.
Banach spaces. Equicontinuity and pointwise compactness.
Arzela-Ascoli Theorem.
Lipschitz condition.
Contraction Mapping Principle and applications to differential and integral equations.
Bernstein polynomials.
Stone-Weierstrass Theorem.
Differentiability and the derivative of vector-valued functions.
Jacobian matrix, gradient vectors.
Differentiable curves and tangent vectors.
Conditions for differentiability.
Directional derivatives. Chain rule. Product rule.

Learning Outcome

solve calculus problems between Euclidean spaces;
examine the type of local extrema of functions;
test the validity of various theorems in analysis;
verify that conditions in analysis are satisfied in order to apply major theorems;
construct matrix representations of the derivative of maps between Euclidean spaces;
construct from definition various types of derivatives of specific maps between Euclidean spaces;
construct proofs of normed, metric, completeness and topological structures, and various properties of given subsets of the space of continuous maps;
create proofs of various properties, identities and relations for continuous, contraction or differentiable mappings;
design iteration schemes that use contraction mappings to solve problems in differential or integral equations.
solve a number of problem sets within strict deadlines;
construct a range of mathematical techniques to solve a variety of formal problems;
formulate solutions to problems individually and/or as part of a team.