1
Banach Space
A normed linear space is a metric space w.r.t the metric d derived from its norm, where d(x, y) =
xy .
Denition A Banach space is a normed space that is a complete.
Example C(K) for K compact, with the sup-norm is a Banach space.
Example C
PDE and Functional Analysis
Winter term 2014/2015
Problem Sheet 13 (due Friday 23.01.2015)
University of Bonn
Prof. Dr. J. J. L. Velzquez
a
Dr. M. Helmers
Problem 1 (2+1 points)
1. Give an example of an open set Rn and a function u W 1, () such that u is
PDE and Functional Analysis
University of Bonn
Winter term 2014/2015
Prof. Dr. J. J. L. Velzquez
a
Problem Sheet 14 (due Friday 30.01.2015 if you want it marked)
Dr. M. Helmers
Examination Information
The rst exam is Monday, 23.02.2015, 912 in the big le
PDE and Functional Analysis
Winter term 2014/2015
Problem Sheet 12 (due Friday 16.01.2015)
University of Bonn
Prof. Dr. J. J. L. Velzquez
a
Dr. M. Helmers
Important notes:
The nal problem sheet that counts towards the 50% for passing the tutorials is She
PDE and Functional Analysis
Winter term 2014/2015
Problem Sheet 11 (due Friday 09.01.2015)
University of Bonn
Prof. Dr. J. J. L. Velzquez
a
Dr. M. Helmers
Problem 1 (1+2+1 points)
1. Consider the operator T : D(T ) L2 (R), (T u)(x) = u (x) dened on D(T )
PDE and Functional Analysis
Winter term 2014/2015
Problem Sheet 9 (due Friday 12.12.2014)
University of Bonn
Prof. Dr. J. J. L. Velzquez
a
Dr. M. Helmers
Problem 1 (2+2 points)
Let X, Y, Z be real normed spaces and let the product space X Y = cfw_(x, y) :
PDE and Functional Analysis
Winter term 2014/2015
Problem Sheet 10 (due Friday 19.12.2014)
University of Bonn
Prof. Dr. J. J. L. Velzquez
a
Dr. M. Helmers
Problem 1 (1+2 points)
Let X be a Banach space, D X dense, and (xk ) X, x X.
1. Show that xk
x as k
PDE and Functional Analysis
Winter term 2014/2015
Problem Sheet 8 (due Friday 05.12.2014)
University of Bonn
Prof. Dr. J. J. L. Velzquez
a
Dr. M. Helmers
Problem 1 (3+1+1 points)
Let Rn be open and bounded and x f L2 (; Rn ), f = (f1 , . . . , fn ). For u
PDE and Functional Analysis
Winter term 2014/2015
Problem Sheet 7 (due Friday 28.11.2014)
University of Bonn
Prof. Dr. J. J. L. Velzquez
a
Dr. M. Helmers
Problem 1 (3 points)
Let 0 < 1 and Rn be open and bounded. Show that bounded sets in C 0, ()
are prec
PDE and Functional Analysis
Winter term 2014/2015
Problem Sheet 5 (due Friday 14.11.2014)
University of Bonn
Prof. Dr. J. J. L. Velzquez
a
Dr. M. Helmers
Problem 1 (2+1+1 points)
Let Rn be open and u W 1,p (), 1 p .
1. Show that u+ = maxcfw_u, 0 and u = m
PDE and Functional Analysis
Winter term 2014/2015
Problem Sheet 2 (due Friday 24.10.2014)
University of Bonn
Prof. Dr. J. J. L. Velzquez
a
Dr. M. Helmers
Problem 1 (Closed sets in metric spaces, 2+2 points)
Let (X, d) be a metric space. Prove that
1. A se
PDE and Functional Analysis
Winter term 2014/2015
Problem Sheet 6 (due Friday 21.11.2014)
University of Bonn
Prof. Dr. J. J. L. Velzquez
a
Dr. M. Helmers
Problem 1 (4 points)
Let
1
1/2
M=
u C 0 ([0, 1]) :
u(x) dx
u(x) dx = 1 .
1/2
0
Show that M is a conv
PDE and Functional Analysis
Winter term 2014/2015
Problem Sheet 4 (due Friday 07.11.2014)
University of Bonn
Prof. Dr. J. J. L. Velzquez
a
Dr. M. Helmers
Problem 1 (An interpolation inequality, 2 points)
1
Let 1 p1 , p2 < , 0 < < 1, and p such that p = p1
PDE and Functional Analysis
Winter term 2014/2015
Problem Sheet 3 (due Friday 31.10.2014)
University of Bonn
Prof. Dr. J. J. L. Velzquez
a
Dr. M. Helmers
Problem 1 (Convergence in Hilbert spaces, 1+1 points)
Let H be a Hilbert space with inner product (,
PDE and Functional Analysis
Winter term 2014/2015
Problem Sheet 1 (due Friday 17.10.2014)
University of Bonn
Prof. Dr. J. J. L. Velzquez
a
Dr. M. Helmers
Problem 1 (Either-Or Topology, 2 points)
Let X = [1, 1] and
T = cfw_A X : either A does not contain 0
Difference equations are now used in all areas of science: in biology, demography,
ecology, economics, engineering, finance, and physics.
This course is an introduction to discrete dynamical systems. In this course we will
consider the fundamental factors