Many
of
the
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, pointwise versus
uniform convergence of sequences of functions, Riemann integrability
etc.
The
interrelation
does
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.
The
first
term
of Ma 108 covers the following topics in real analysis:
 Ordered
sets,
upper
and
lower bounds, least upper bound axiom, the construction
of real numbers.
 Topology
of
metric
spaces.
Open, closed, bounded and compact sets, perfect
and connected sets. Sequences in metric spaces.
 The
HeineBorel
Theorem.
The
BolzanoWeierstrass Theorem. Compact metric
spaces.
 Normed
linear
spaces.
Completeness.
Banach spaces. Contraction Mapping
Principle and applications.
 Sequences
and
series
of functions. Uniform convergence. Equicontinuity.
The ArzelaAscoli theorem. The Weierstrass approximation theorem.
 Bounded
linear
transformations.
The
derivative as a linear transformation.
Taylor theorem in Rn.
Prerequisites: Calculus
of
one
and several variables.
