Prof. Girardi
13. A Review of some Topology
13.1. Definition. Let X be a non-empty set. Let P(X) be the collection of all subsets of
X, i.e. P(X) is the power set of X. Let τ ⊂ P(X). Then (X, τ ) is a topological space (or
equivalently τ is a topology on X) provided
• if A, B ∈ τ , then A ∩ B ∈ τ
• for any index set Γ, if {Aα }α∈Γ ⊂ τ then ∪α∈Γ Aα ∈ τ
• ∅, X ∈ τ .
In which case, A ∈ P(X) is an open set if and only if A ∈ τ .
13.2. Definition. Let (X, τ ) be a topological space and B ⊂ τ . Then B is a base (or basis)
of τ provided
∀A ∈ τ ∃{Bα }α∈Γ ⊂ τ such that A = ∪α∈Γ Bα .
13.3. Example. Let X be a normed linear space. As usual, for ε > 0 and x ∈ X, let
Nε (x) := {y ∈ X : kx − ykX < ε} .
Let τ be the open subsets of X, i.e.,
τ = {A ∈ P(X) : ∀x ∈ A, ∃ε > 0 such that Nε (x) ⊂ A} .
Then a base B for (X, τ ) is
B := {Nε (x) : x ∈ X, ε > 0} .
13.4. Recall. Read through the course handouts, which are posted on the course homepage,
for Section 13.
13.5. Definition. Let X be a vector space over the field K where, as usual, K is R or C.
• A linear functional on X is a linear mapping f : X → K.
• Let Z be a family of linear functionals on X. Then Z separates points provided if
x, y ∈ X and x 6= y then there is a f ∈ Z such that f (x) 6= f (y).
• Let (X, τ ) be a topological space.
The topological dual of X, denoted by (X, τ )∗ , is the set of all continuous linear
functionals on X, i.e.,
(X, τ )∗ := {f : X → K | f is continuous and linear} .
13.6. Definition/Recall. Let Z be a family of linear functionals that separates points on
a vector space X.
Then σ(X, Z) denotes the weakest (i.e. smallest) topology on X for which each f ∈ Z is
continuous.
• Clearly, each f ∈ Z is continuous with respect to the (large) topology σ(X, P(X))
on X.
• If τ 0 is any topology on X such that each f ∈ Z is continuous with respect to (X, τ 0 )
then σ(X, Z) ⊂ τ 0 .
A base for σ(X, Z) is
n −1
∩i=1 fi (Ui ) : fi ∈ Z , Ui is open in K , n ∈ N .
Another, more convenient, base for σ(X, Z) is
n −1
∩i=1 fi [Nε (fi (x0 ))] : fi ∈ Z , ε > 0 , x0 ∈ X , n ∈ N ;
note that
∩ni=1 fi−1 [Nε (fi (x0 ))] = {x ∈ X : |fi (x) − fi (x0 )| < ε , i = 1, . . . , n}
= {x ∈ X : |fi (x − x0 )| < ε , i = 1, . . . , n} =: Nε,f1 ,...,fn (x0 ) .
Since Z separates points, σ(X, Z) is a Hausdorff (i.e., T2 ) topology, i.e., if x1 , x2 ∈ X and
x1 6= x2 then there are disjoint U1 , U2 ∈ σ(X, Z) such that xi ∈ Ui . Thus, if the limit of a
net from X exists, then the limit is unique.
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Prof. Girardi
13. A Review of some Topology
13.7. Fact. Let X be a vector space. Let Z be a vector space of linear functionals on X that
separates points. Then
(X, σ (X, Z))∗ = Z .
That (X, σ (X, Z))∗ ⊃ Z follows from the definition. And ⊂ follows from linear algebra
(see [RS-I, Thm. IV.20]).
13.8. Optional Example. Let X be a Banach space over K, endowed with a norm k·kX and
the norm topology τk·kX as in Example 13.3. Let
∗
X ∗ := X, τk·kX
= {x∗ : X → K | x∗ is linear and (norm) continuous} .
(1) The σ(X, X ∗ ) topology on X is called the weak topology.
A basis for the weak topology consists of all sets of the form
Nε,x∗1 ,...,x∗n (x) = {y ∈ X : |x∗i (x − y)| < ε , for i = 1, . . . , n} .
where x ∈ X and x∗1 , . . . , x∗n ∈ X ∗ and ε > 0 and n ∈ N.
ε
Note σ(X, X ∗ ) ⊂ τk·kX since Nε,x∗1 ,...,x∗n (x) ⊃ N
(x).
max kx∗i k
By Fact 13.7, (X, σ (X, X ∗ ))∗ = X ∗ . So for a linear functional x∗ : X → K, x∗ weak
continuous if and only if x∗ is norm continuous.
(2) The σ(X ∗ , X) topology on X ∗ is called the weak∗ topology.
A basis for the weak∗ topology consists of all sets of the form
Nε,x1 ...,xn (x∗ ) = {y ∗ ∈ X ∗ : |(x∗ − y ∗ )(xi )| < ε , for i = 1, . . . , n} .
where x∗ ∈ X ∗ and x1 , . . . , xn ∈ X and ε > 0 and n ∈ N.
Note σ(X ∗ , X) ⊂ σ(X ∗ , X ∗∗ ) since X ⊂ X ∗∗ .
By Fact 13.7, (X ∗ , σ (X ∗ , X))∗ = X. So for a linear functional x∗∗ : X ∗ → K, x∗
weak∗ continuous if and only if x∗∗ comes from X, i.e, there is x ∈ X such that
x∗∗ (x∗ ) = x∗ (x) for each x∗ ∈ X ∗ .
13.9. Optional Fact. Let X be a normed linear space. Then the following are equivalent.
• dim X < ∞.
• (X, σ (X, X ∗ )) is metrizable.
• (X ∗ , σ (X ∗ , X)) is metrizable.
Thus, often sequences do not suffice and we need to consider nets. (See course handouts on
nets.)
13.10. Definition. Let X be a vector space over K. Let ρ : X → [0, ∞) be a function.
Then ρ is a seminorm (in German, Halbnorm) provided ∀x, y ∈ X and α ∈ K:
• ρ(x + y) ≤ ρ(x) + ρ(y)
• ρ(α x) = |α| ρ(x) .
Let’s think about this definition. Why might a seminorm not be a norm? Well, because
ρ(x) = 0 need not imply that x = 0. So we cope with this badness by the following definition.
A family {ρα }α∈A of seminorms separates points provided if ρα (x) = 0 for each α ∈ A
then x = 0.
13.11. Definition. A space X is called a locally convex topological vector space (i.e. (LCTVsp))
provided
(1) X is a vector space over K
(2) ∃ a family {ρα }α∈A of seminorms on X that separates points.
In this case:
(3) the natural topology τ = τ{ρα }α∈A on X, generated by the seminorms {ρα }α∈A , is the
weakest topology so that:
• ρα is continuous for each α ∈ A
• the mapping + : X × X → X, defined by (x, y) → x + y, is continuous .
Note that since {ρα }α∈A separates points, τ is Hausdorff.
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13. A Review of some Topology
(4) a basis of τ is
{Nε,α1 ,...,αn (x) | αi ∈ A , ε > 0 , n ∈ N , x ∈ X}
where Nε,α1 ,...,αn (x) := {y ∈ X : ραi (x − y) < ε , i ∈ {1, . . . , n}}, which is a convex
set
(5) if {xβ }β∈B is a net in X, then
xβ → x
⇐⇒
ρα (xβ − x) → 0 ∀α ∈ A .
13.12. Example. Let X be a Banach space. The natural topology generated by the family
of seminorms consisting of:
• just one seminorm, namely the norm on X, is just the usual norm topology on X.
• {ρx∗ (·) := |x∗ (·)|}x∗ ∈X ∗ is weak topology σ(X, X ∗ ) on X from Example 13.8.
13.13. Definition. Let X, τ{ρα }α∈A be a LCTVsp. Let {xβ }β∈B be a net in X.
• {xβ } is Cauchy if and only if [∀ε > 0 , ∀α ∈ A , ∃βε,α ∈ B such that if β, γ >
βε,α then ρα (xβ − xγ ) < ε .
• X is complete if and only if [each Cauchy net in X converges to a point in X].
13.14. Proposition. Let {ρα }α∈A and {dβ }β∈B be 2 families of seminorms on X. Then the
following are equivalent.
(1) {ρα }α∈A and {dβ }β∈B are equivalent families of seminorms, which by definition
means that τ{ρα }α∈A = τ{dβ }β∈B .
(2) ∀α ∈ A , ρα is τ{dβ }β∈B -continuous
and
∀β ∈ B , dβ is τ{ρα }α∈A -continuous .
(3) ∀α ∈ A , ∃β1 , . . . , βn ∈ B , ∃C > 0 , such that ∀x ∈ X , ρα (x) ≤ C(dβ1 (x) + . . . + dβn (x))
and ∀β ∈ B , ∃α1 , . . . , αn ∈ A , ∃D > 0 , such that ∀x ∈ X , dβ (x) ≤ D(ρα1 (x) + . . . + ραn (x)) .
13.15. Definition. A family {ρα }α∈A of seminorms on X is directed provided
∀α, β ∈ A , ∃γ ∈ A , ∃C > 0 , such that ∀x ∈ X : ρα (x) + ρβ (x) ≤ Cργ (x) .
Note that, by induction, this is equivalent to
∀α1 , . . . , αn ∈ A , ∃γ ∈ A , ∃D > 0 , such that ∀x ∈ X : ρα1 (x) + . . . + ραn (x) ≤ Dργ (x) .
13.16. Remark. Let {ρα }α∈A be a family of seminorms on X. Define
• B = the set of all finite
Psubsets of A
• for F ∈ B define dF := α∈F ρα .
Then
• {dF }F ∈B is a directed family of seminorms on X
• τ{ρα }α∈A = τ{dF }F ∈B .
Recall. A linear map T from a normed space X to a normed space Y is continuous if and only
if there exists C > 0 so that kT xkY ≤ C kxkX for each x ∈ X. The next proposition is the
corresponding statement for maps between spaces with topologies generated by seminorms.
13.17. Proposition. Let T : X, τ{ρα }α∈A → Y, τ{dβ }β∈B be a linear map. The following
are equivalent.
(1) T is continuous
(2) ∀β ∈ B , ∃α1 , . . . , αn ∈ A , ∃C1 > 0 such that ∀x ∈ X : dβ (T x) ≤ C1 (ρα1 (x) + . . . ραn (x))
(3) ∀β ∈ B , ∃ a continuous seminorm ρ on X and ∃C2 > 0 such that ∀x ∈ X : dβ (T x) ≤
C2 ρ(x)
And if {ρα }α∈A is a directed set, then also
(4) ∀β ∈ B , ∃α ∈ A , ∃C3 > 0 such that ∀x ∈ X : dβ (T x) ≤ C1 ρα (x) .
13.18. Remark.
We will often
use Proposition 13.17 in the case of the formal identity j :
X, τ{ρα }α∈A → X, τk·kX . In this case, j is continuous if and only if ∃α1 , . . . , αn ∈
A , ∃C > 0 such that ∀x ∈ X : kT xkX ≤ C (ρα1 (x) + . . . ραn (x)).
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13. A Review of some Topology
13.19. Proposition. Let X be a LCTVsp with a natural topology τ generated by a family
of seminorms. The following are equivalent.
(1) (X, τ ) is metrizable, i.e., ∃ a metric d : X × X → [0, ∞) such that τ equals the
topology generated by the metric d.
(2) ∃ a countable family {ρn }n∈N of seminorms on X that separates points such that
τ = τ{ρn }n∈N .
To show that (2) ⇒ (1), let
X
ρn (x − y)
−n
.
(1)
d(x, y) :=
2
1 + ρn (x − y)
n∈N
In which case, it easy to see that if {xα }α∈A is a net in X, then
[{xα }α∈A is Cauchy in the metric d in (1)] ⇔ [{xα }α∈A is Cauchy for each seminorm ρn ] .
Thus a metrizable LCTVsp is complete as a metric space if and only if it is complete as a
LCTVsp.
13.20. Definition.
1
A Fréchet space is a complete metrizable LCTVsp.
13.21. Theorem (Open Mapping Theorem). Let X and Y be Fréchet spaces. Let f : X → Y
be a continuous linear surjective mapping. Then f is open, i.e., if A is an open set in X
then f (A) is an open set in Y .
13.22. Corollary. Let X and Y be Fréchet spaces. Let f : X → Y be a continuous linear
bijective mapping. Then f is a topological homeomorphism, i.e., f is bijective and f and
f −1 are continuous.
13.23. Theorem. Let X be a Fréchet space. Let f, fn ∈ X ∗ be so that {fn }n∈N converges to
f in the σ(X ∗ , X)-topology. Then {fn }n∈N converges to f uniformly on compact subsets of
X, i.e.,
∀ compact D ⊂ X , ∀ε > 0 , ∃N ∈ N , such that ∀n ≥ N , sup |fn (x) − f (x)| < ε .
x∈D
References
[RS-I] Michael Reed and Barry Simon, Methods of modern mathematical physics I, Second edition. Academic
Press Inc., New York, 1980.
1This
definition varies from book to book.
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