1. Let k · k0 and k · k00 be two norms on a vector space X. Prove that they are equivalent
if and only if the identity operator (X, k · k0 ) → (X, k · k00 ), x 7→ x, is an isomorphism.
2. Prove that c and c0 are isomorphic.
3. Show that c and c0 are not isometrically isomorphic by computing extremal points of
the unit balls of these spaces.
If K is a convex subset of a vector space X, a point x ∈ K is called extremal if the only
way to write x as λy + (1 − λ)z for some λ ∈ (0, 1) and y, z ∈ K is to have y = z = x.
4. Prove the following Mazur theorem. If K is a compact subset of a Banach space X
then the closed convex hull co K of K is also compact.
By definition
co K is the closure of the set of points of the form λ1 x1 + . . . + λn xn , where
P
λk ≥ 0, k λk = 1, xk ∈ K, n ∈ N.
5. Let K be a compact space. A subset M ⊂ C(K) is called equicontinuous if for any
x ∈ K and ε > 0 there exists a neighbourhood U of x such that |f (y) − f (x)| < ε for any
f ∈ M and y ∈ U . Show that if K is a compact metric space and M is equicontinuous then
for any ε > 0 there exists δ > 0 such that |f (y) − f (x)| < ε for any f ∈ M and x, y ∈ K
such that d(x, y) < δ.