Problem solving seminar VI
21. Let U ⊆ Rn
be an open set. Suppose that the map h : U → Rn
is a homeomorphism
from U onto Rn
, which is uniformly continuous. Prove that U = Rn
.
22. Suppose that f maps the compact interval I into itself and that
|f(x) − f(y)| < |x − y|
for all x, y ∈ I, x = y. Can one conclude that there is some constant M < 1 such that
|f(x) − f(y)| < M|x − y|?
23. Let V be a finite dimensional vector space and A and B two linear transformations
of V into itself such that A2
= B2
= 0 and AB + BA = id.
(a) Prove that ker A = A(ker B), ker B = B(ker A) and V = ker A ⊕ ker B.
(b) Prove that the dimension of V is even.
24. Prove that the group G = Q/Z has no proper subgroup of finite index.
Homework VI.
(a) Let M be a compact metric space and let f : M → R be an upper semicontinuous
function. Prove that f is bounded from above and that it takes its maximum in a
point of M.
(b) Let I and J be two metric spaces and let g : I×J → R be continuous and bounded
from below. Prove that the function f : I → R defined
f(x) = inf{g(x, y); y ∈ J}
is upper semicontinuous.
1