1. Prove that c∗ is isometrically isomorphic to `1 .
2. Prove that if the unit ball in a normed space is compact then the space is finite
dimensional.
3. Let {Xi }i∈I be a collection of normed spaces, and 1 ≤ p < ∞. Consider the set
M
p Xi
i∈I
consisting of sets {xi }i∈I such that xi ∈ Xi and
X
kxi kp < ∞.
i
Prove that this is a vector space and
!1/p
k{xi }i∈I k =
X
kxi kp
i
is a norm on it.
4. Let Y be a closed subspace of a normed space X. Show that there exist isometric
isomorphisms
Y∗ ∼
= X ∗ /Y ⊥ and (X/Y )∗ ∼
= Y ⊥.
5. Show that the space C[a, b] is not reflexive.
6. Let Y be a subspace of a normed space X. Show that Y ∗∗ can be identified with
(Y ⊥ )⊥ ⊂ X ∗∗ .
Conclude that if X is a reflexive Banach space then any closed subspace of X is also
reflexive.
7. Let L be a Banach limit, that is, a linear functional on `∞ such that
(i) kLk = 1;
(ii) L(a) = limn an for a ∈ c;
(iii) if an ≥ 0 for any n then L(a) ≥ 0;
(iv) if a and a0 are such that a0n = an+1 then L(a) = L(a0 ).
Compute L((1, 0, 1, 0, . . .)).