Mathematics 442C
Exercise sheet 3
Due 12pm, Thursday 12th November 2009
1. Prove Remark 2.2.3:
Suppose that T is a topology on a set X, let S ⊆ T , and let
Sb =
ni
n[ \
Sij : I is a set, ni ≥ 0 and Sij ∈ S for i ∈ I and 1 ≤ j ≤ ni
o
i∈I j=1
be the collection of unions of finite intersections of sets in S.
(a) Show that Sb ⊆ T and that Sb is a topology on X.
(b) Show that the following conditions are equivalent:
(i). S is a subbase for T ;
(ii). Sb = T ;
(iii). T is the collection of sets G ⊆ X such that for every x ∈ G, there
are finitely many sets S1 , . . . , Sn ∈ S such that x ∈ S1 ∩· · ·∩Sn ⊆ G.
(c) Suppose that Y is another topological space. Show that a function
f : Y → X is continuous if and only if f −1 (S) is open in Y , for all
S ∈ S.
2. Let (X, d) be a metric space and let T be the topology arising from the
metric d (as in [FA 1.2.3]).
(a) For x ∈ X and ε > 0, let B(x, ε) = {y ∈ X : d(x, y) < ε}. Show that
{B(x, ε) : x ∈ X, ε > 0}
is a subbase for T .
(b) For x ∈ X, let dx : X → R, y 7→ d(x, y). Show that T is the weak
topology induced by the family {dx : x ∈ X}.
(c) Deduce that T is the weak topology induced by C(X, T ), the set of
functions X → C which are continuous with respect to the topology T .
3. Let X be a Banach space and for x1 , . . . , xn ∈ X, ϕ ∈ X ∗ and ε > 0, let
U (ϕ; x1 , . . . , xn ; ε) = {ψ ∈ X ∗ : |ψ(xi ) − ϕ(xi )| < ε for 1 ≤ i ≤ n}.
(a) Use (one of the corollaries of) the Hahn-Banach theorem to show that if
x ∈ X with x 6= 0 and z ∈ C then there is some ϕ ∈ X ∗ with ϕ(x) = z.
(b) For z ∈ C and ε > 0, let B(z, ε) = {w ∈ C : |w − z| < ε}. Show that if
x ∈ X then the map Jx : X ∗ → C, ψ 7→ ψ(x) satisfies
Jx−1 (B(ϕ(x), ε)) = U (ϕ; x; ε).
(c) Show that S = {U (ϕ; x; ε) : ϕ ∈ X ∗ , x ∈ X, ε > 0} is a subbase for the
weak* topology on X ∗ .
[Hint: the sets B(z, ε) form a subbase for the topology on C [why?].
Use this fact with Exercise 1(c) and part (b).]
(d) Show that if ψ ∈ U (ϕ; x; ε) where ϕ ∈ X ∗ , x ∈ X and ε > 0, then there
is some δ > 0 such that U (ψ; x; δ) ⊆ U (ϕ; x; ε).
(e) Show that a set G ⊆ X ∗ is open in the weak* topology if and only if, for
every ϕ ∈ G, there are finitely many points x1 , . . . , xn ∈ X and ε > 0
such that U (ϕ; x1 , . . . , xn ; ε) ⊆ G.
4. Consider the Banach sequence spaces
ℓ∞ = {(xn )n≥1 : xn ∈ C, sup |xn | < ∞} and
n≥1
X
ℓ1 = {(yn )n≥1 : yn ∈ C,
|yn | < ∞}.
n≥1
Recall from [FA 3.4] that ℓ∞ is the dual space
of ℓ1 , where for x = (xn )n≥1 ∈
P
ℓ∞ and y = (yn )n≥1 ∈ ℓ1 we have x(y) = n≥1 xn yn .
For n ≥ 1, let en ∈ ℓ∞ be the sequence with 1 in the nth position and
zeros elsewhere.
(a) Show that K = {en : n ≥ 1} is a norm-closed subset of ℓ∞ .
(b) Show that en (x) → 0 as n → ∞ for any x ∈ ℓ1 .
(c) Show that, in the weak* topology on ℓ∞ , we have en → 0 as n → ∞.
(d) Deduce that the weak* topology on ℓ∞ is strictly weaker than the norm
topology on ℓ∞ .
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