Mid-Sem Functional Analysis
Each question is worth 5 points. Write clearly.
1 2
1. Let L : R → R be the linear transformation given by
in terms of the standard basis
3 4
2
of R . Find ||L||1 and ||L||∞ . What can you say about ||L||p where p ∈ (1, ∞). (5 pts)
2
2
2. Let C[0, 1] be the space of real-valued continous functions on [0, 1] equipped with sup norm.
a) Show that this norm does not come from any inner product. (5 pts)
b) Consider the integral operator
K : C[0, 1] →
(1)
C[0, 1]
Z
f (x) 7→
F (s) =
s
f (x)dx.
(2)
0
Show that K is bounded. Compute the norm of ||K||. (5 pts)
3. Recall that definition of `∞ and the subspace c00 ⊂ `∞ . Is c00 a Banach space? Describe the
closure of c00 in `∞ . (5 pts)
1