Department of Mathematics
MATH4010 Functional Analysis
Homework 7
Notice:
• All the assignments must be submitted before the deadline.
• Each assignment should include your name and student ID number.
1. Let S be a bounded sesquilinear form on X × Y . Define
kSk := sup {|S(x, y)| : kxk = 1, kyk = 1} .
Show that
kSk = sup
|S(x, y)|
kxkkyk
: x ∈ X \ {0}, y ∈ Y \ {0}
and
|S(x, y)| ≤ kSkkxkkyk,
for all x ∈ X and y ∈ Y .
2. Let T : `
2 → `
2 be defined by
T : (x1, . . . , xn, . . .) 7→ (x1, . . . ,
1
n
xn, . . .).
Show that the range R(T) is not closed in `
2
.
3. Let T be a bounded operator on a complex Hilbert space H.
(a) Show that the operators
T1 =
1
2
(T + T
∗
) and T2 =
1
2i
(T − T
∗
)
are self-adjoint.
(b) Show that T is normal if and only if the operators T1 and T2 commute.
— THE END —
1
