MATH 7280 Operator Theory-Spring 2006
Instructor: Marian Bocea
Homework Assignment # 1
(Due February 17, 2006)
1. Let E be a normed space, and f : E → R be a linear map. Show that the
hyperplane
[f = α] := {x ∈ E : f (x) = α}
is closed if and only if f is continuous.
2. Let c0 := {{xn }n∈N : xn → 0 as n → ∞} be the vector space of sequences of
real numbers converging to 0 endowed with the norm
k{xn }n∈N k := sup |xn |.
N
n∈
Consider f : c0 → R given by f ({xn }n∈N ) :=
Show that
∞
P
n=1
xn
.
n2
(i) f is linear and continuous
(ii) the supremum in
kf kc00 = sup {hf, xi : x ∈ c0 , kxk ≤ 1}
is not attained.
3. Give an example of two linear continuous maps f1 , f2 : c0 → R, f1 6= f2 , and an
element x0 ∈ c0 such that
kf1 kc00 = kf2 kc00 = kx0 k and hf1 , x0 i = hf2 , x0 i = kx0 k2 .
1