Math 595 --Transition course for graduate studies

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.


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

Charles W. Curtis:  Linear Algebra-An Introductory approach,
Springer 1984, Sections 2 and 7.

Homework (30%)-individual submission (always Mondays), projects (10%)
2 Midterm exams  (30%)

Final exam  (30%)














solutions to final partice

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