Mathematics 414 2003–04
Exercises 6
[Due Monday March 1st, 2004.]
1. Let (fn )∞
n=1 be a sequence in C(G) (where G ⊂ C is open), f : G → C a function and
K ⊂ G compact. Verify carefully that fn → f uniformly on K (as n → ∞) if and only if
dK (fn , f ) = sup |fn (z) − f (z)| → 0
z∈K
as n → ∞.
2. For each of the following open G ⊂ C, exhibit explicitly an exhaustive sequence (Kn )∞
n=1
of compact subsets of G.
(a) G = D(a, r)
(b) G = {z ∈ C : r1 < |z| < r2 }
(c) G = C \ Z
(d) G = {z ∈ C : 0 < <z < 1}
3. Let G ⊂ C be open, (fn )∞
n=1 a sequence of functions fn : G → C and f : G → C a function.
We say that fn → f locally uniformly on G if for each z0 ∈ G there is a disk D(z0 , δ)
about z0 (of positive radius δ > 0) so that fn → f uniformly on D(z0 , δ).
Show that fn → f locally uniformly if and only if fn → f uniformly on compact subsets
of G.
4. Let G ⊂ C be open and ρ a metric on H(G) such that convergence in ρ corresponds to
uniform convergence on compact subsets of G. Fix z0 ∈ G. Show that that the point
evaluation map δz0 : H(G) → C given by δz0 (f ) = f (z0 ) is continuous.
That is show that if fn → f in (H(G), ρ) then δz0 (fn ) → δz0 (f ).
Show that δz0 is linear and multiplicative (δz0 (f g) = δz0 (f )δz0 (g)). Deduce that I =
ker δz0 = {f ∈ H(G) : δz0 (f ) = 0} is a closed ideal in H(G) (that is I is a vector
subspace, has the property f g ∈ I if f ∈ I, g ∈ H(G) and is also closed as a subset of
H(G)).
5. Let G ⊂ C be open and ρ a metric on H(G) such that convergence in ρ corresponds to
uniform convergence on compact subsets of G.
(a) Show that the map f 7→ f 0 : H(G) → H(G) is continuous.
That is show that if fn → f in (H(G), ρ) then fn0 → f 0 in (H(G), ρ). [Hint: Use the
Cauchy integral formula to show local uniform convergence.]
(b) Show that if F ⊂ H(G) is relatively compact, then {f 0 : f ∈ F} is also relatively
compact in H(G).