Download Homework 6 (1) Let φ : G → C be a function of positive type on the

Tools of Modern Analysis. Winter 2014.
Homework 6
(1) Let φ : G → C be a function of positive type on the group G.
Show that the function x 7→ φ(gx) can be written as a linear
combination of functions of positive type.
(2) Let φ be a positive definite function on the group G. Show that
H = {g ∈ G : φ(g) = φ(e)}
is a subgroup of G.
(3) (a) Show that every open subgroup of a topological group is
closed.
(b) Let G be a connected topological group and U any open
subset. Prove that U generates G.
(4) Write formulas for left and right invariant Haar measures on
the group
a b
: a > 0, b ∈ R .
0 1
(5) Let µ denote the Haar measure on a topological group. Prove
that for every open set U , one has µ(U ) > 0.
(6) Prove that if the Haar measure on a topological group is finite.
Then the group is compact.
(7) Let G be a compact group that acts continuously on a compact
metric space. Show that there exists a metric which defines
the same topology and is G-invariant. (Hint: use the Haar
measure.)
1