**Time:** MWF 1-2 pm

**Location: ** Altgeld
Hall 341

**Instructor:** Marius
Junge Course email
Office hours: Wednesday 2-3 or by appointment

Intention of the course
: We have realized that graduate students with very different
backgrounds come to our university. This concerns in particular the
ability to work with abstract concepts, formal proofs and basic
knowledge in analysis and linear algebra

(which might formally qualify as 'undergraduate material'). This course
is not a review course. To the contrary we will treat interesting,

but fundamental material, on a level which is
appropriate for graduate students in pace and complexity. We will also
encourage projects where students are encouraged to fill gaps in their
knowledge by independent research (also in peer team work). A
particular focus of this course is the interaction of analytic and
algebraic concepts.

Course discription
:

Part I: Metric spaces
(script including compact spaces)

1) definition of
metric spaces, space
of continuous functions,

2) complete metric spaces, existence of the
completion, unique extension of continuous functions, three proofs

3) compactness, equivalent conditions
(sequentially compact, totally bounded and complete), continuous

functions attain
their maximum, Heine-Borel Theorem,

4) contraction
mapping principle with
application to Picard-iteration,

5) definition of topological spaces and connected
sets.

Part II: Vector
spaces and topological properties

1) definition of vectors spaces over R and C, linear
maps and spaces of linear maps,

2) basic properties of the minimal polynomial (in comparison
with the characteristic polynomial),

3) eigenvector and generalized eigenspaces for linear maps,
matrices and change of basis, characterization of diagonalizable maps
in terms of

the minimal polynomial,

4) Discussion of the Jordan normal form of a linear map and the
form of the blocks, sketch of proof,

5) the
definition of topological vector spaces, in
particular normed linear spaces, the space of continuous linear
maps and completeness,
characterization of completeness in terms of absolutely convergent
series, uniform convergence of powe series,

6) differentiable functions between normed linear
spaces, differentiation of power series, solution of
f'(t)=A(f(t)) for bounded linear maps A,
calculating e^{tA}, cos(A) and sin(A)
(1-A)^{-1} using the Jordan normal form, applications
to systems
of linear differential equations,

7) proof
of the inverse function theorem.

Part III: Elementary Geometry in Hilbert spaces

1) the scalar product and the Cauchy-Schwarz inequality,
parallelogram equality,

2) characterization of minima of convex functions in terms of
directional derivatives,

3) applications to least norm approximations, the existence of
orthogonal projections,

4) existence of orthogonal basis, easy version of Bessel's
inequality, illustration for basis of eigenvalues for selfadjoint
matrices.

Books:

Juergen Jost: Postmodern Analysis, Springer 1988,Sections 6-10.Grading:

Charles W. Curtis: Linear Algebra-An Introductory approach,

Springer 1984, Sections 2 and 7.

Homework (30%)-individual submission (always Mondays), projects (10%)hw4

2 Midterm exams (30%)

Final exam (30%)

hw5

takehome

hw6

hw7

hw8

hw9

hw10

hw11

grades

solutions to final partice