results encountered in calculus seem to follow common themes,
even though they are usually formulated as different topics. As an
example think of the various notions of convergence typically discussed
in a calculus class: limits of a sequence of numbers, point-wise versus
uniform convergence of sequences of functions, Riemann integrability
become very clear when viewing the objects under
consideration as elements of more abstract spaces equipped with some
``notion of closeness'' (topology). Goal of this class is to develop
some of the frameworks (``metric spaces'') allowing for this more
general point of view, which will naturally lead to many of the
questions of classical real analysis.
of Ma 108 covers the following topics in real analysis:
lower bounds, least upper bound axiom, the construction
of real numbers.
Open, closed, bounded and compact sets, perfect
and connected sets. Sequences in metric spaces.
Bolzano-Weierstrass Theorem. Compact metric
Banach spaces. Contraction Mapping
Principle and applications.
of functions. Uniform convergence. Equicontinuity.
The Arzela-Ascoli theorem. The Weierstrass approximation theorem.
derivative as a linear transformation.
Taylor theorem in Rn.
and several variables.