Exercise Sheet 2 in
Functional Analysis,
LMU Munich, Summer Semester 2017
Peter Philip, Sabine Bögli
May 4, 2017
Hand in until 2 P.M. sharp on Monday, May 15 (locked homework box near the library).
1. (10 points) Let X be a vector space over K equipped with the indiscrete/trivial topology
T = {∅, X} (which turns X into a topological vector space, c.f. Ex. 1.2(b)).
(a) Show that every non-empty subset of X is dense in X.
(b) Let A : X → K be a K-linear functional. Prove that A is continuous if and only if
A = 0.
2. (10 points)
(a) Consider an arbitrary topological vector space (X, TX ) over K, and equip Y :=
KR = F(R, K), i.e. the set of functions f : R → K, with the product topology TY
((Y, TY ) is a topological vector space by Ex. 1.2(d)). Prove that a K-linear function
A : X → Y is continuous if and only if the composition πx A : X → K is continuous
for every x ∈ R.
(b) Let X be the space of all real-valued
PN where
P polynomials on R with norm topology
n
the norm is defined by kpk := N
|a
|
for
the
polynomial
p(x)
=
n
n=0 an x .
n=0
Check if the following functionals Ai : X → R are continuous:
Z 1
A1 p :=
p(x) dx, A2 p := p0 (0), A3 p := p0 (1).
0
3. (10 points) Let D := {z ∈ C : |z| < 1} denote the unit disc and let X be the space of
continuous functions f : D → C with the topology of uniform convergence on compact
subsets, for which a subbase is given by
(
)
(1)
g + f ∈ X : sup |f (z)| < : δ ∈ [0, 1), ∈ R+ , g ∈ X .
|z|≤δ
Show that (X, T ) is metrizable but not normable.
Exercise 4 on the back!
4. (10 points) Prove the following
Lemma.
(a) Let X be a vector space and let d be a translation-invariant metric on X (here,
we do not assume that d makes X into a topological vector space). Then
∀
x∈X
∀
n∈N
d(nx, 0) ≤ nd(x, 0).
(b) Let (X, T ) be a metrizable topological vector space over K and let (xn )n∈N be a
sequence in X such that limn→∞ xn = 0. Then there exists an increasing sequence
(Nn )n∈N in N such that limn→∞ Nn = ∞ and limn→∞ Nn xn = 0.