Download All the topological spaces are Hausdorff spaces and all the maps

All the topological spaces are Hausdorff spaces and all the maps are assumed to be continuous.
1. Lecture 1: Continuous family of topological vector spaces
Let k be the field of real numbers R or the field of complex numbers C.
Definition 1.1. A topological vector space over k is a k-vector space V together with a topology
such that the two functions α, β defined below are continuous.
(1) α : V × V → V is the function of the vector addition
α : V × V → V defined by (v, w) 7→ v + w,
(2) β is the function of the scalar multiplication
β : k × V → V defined by (a, v) 7→ av.
Here the two spaces V × V and k × V are equipped with the product topologies.
Example 1.1. We equip k n with the topology defined by the Euclidean norm
p
kak = |a1 |2 + · · · + |an |2
for any a = (a1 , · · · , an ) ∈ k n . This topology is called the Euclidean topology on k n . One can see
that k n is a topological vector space with respect to the Euclidean topology.
Definition 1.2. A morphism from a topological k-vector space V to another topological k− vector
space W is a continuous linear map T : V → W. The set of all morphisms from V to W is denoted
by Hom(V, W ).
Lemma 1.1. Let V and W be any topological k-vector spaces. The set Hom(V, W ) of all morphisms
from V to W forms a vector subspace of the space of k-linear maps Homk (V, W ).
Proof. Let f, g : V → W be continuous linear maps. Let α : W × W → W be the vector addition
function on W. Since W is a topological vector space, α is continuous. For v ∈ V,
(f + g)(v) = f (v) + g(v) = α(f (v), g(v)).
We see that f + g : V → W is the composition of functions:
(f,g)
α
f + g : V −−−−→ W × W −−−−→ W
Here (f, g) : V → W ×W is the function sending v to (f (v), g(v)). Since f, g : V → W are continuous,
the function (f, g) : V → W × W is continuous. Since the composition of continuous functions is
continuous, f + g = α(f, g) is continuous.
Let a ∈ k. Since β : k × W → W is continuous, the restriction of β to {a} × W is continuous.
Since the function V → {a} × W sending v to (a, f (v)) is continuous, the composition af = β(a, f )
is continuous.
Let X be a topological space. A continuous family of (topological) k-vector spaces over X is a
space E together with a continuous surjective map p : E → X such that the set Ex = p−1 (x) has a
structure of finite dimensional k-vector space for each x ∈ X and that the subspace topology on Ex
induced from E makes Ex a topological vector space isomorphic to k n for some n in the category of
topological vector spaces. The map p is called the projection map and E is called the total space of
the family, and the space X is called the base space of the family and Ex is called the fiber over x.
A continuous family of vector spaces p : E → X over X is denoted by (E, p, X).
Example 1.2. Let X be a topological space and V be a finite dimensional normed vector space
over k. Let X × V be the Cartesian product of X and V. Equip X × V with the product topology.
Let p : X × V → X be the projection map (x, v) 7→ x. Then p is continuous (by definition). The
fiber Ex over x is the set Ex = {(x, v) ∈ E : v ∈ V }. For (x, v) and (x, w) in Ex and a ∈ k, we define
(x, v) +x (x, w) = (x, v + w),
1
a ·x (x, v) = (x, av).
2
Then Ex becomes a k-vector space that is isomorphic to V. One can check that (X × V, p, X) is a
continuous family of k-vector spaces over X. The family (X × V, p, X) called a product family over
X.
A morphism ϕ from a family 1 (E, p, X) to a family (E 0 , p0 , X) is a continuous map ϕ : E → E 0
such that p0 = ϕ◦p. The set of all morphisms from (E, p, X) to (E 0 , p0 , X) is denoted by Hom(E, E 0 ).
An isomorphism ϕ from (E, p, X) to (E 0 , p0 , X) is a morphism such that ϕ : E → E 0 is a bijection
whose inverse ϕ−1 : E 0 → E determines a morphism from (E 0 , p0 , X 0 ) to (E, p, X).
Definition 1.3. A family (E, p, X) over X is said to be trivial if it is isomorphic to a product family
over X. If the family p : E → X is a trivial family and V is a finite dimensional normed vector space
over k so that E is isomorphic to X × V, an isomorphism h : E → X × V is called a trivialization
of E over X modeled on V.
Example 1.3. Let Y be a subspace of X and p : E → X be a family over X. Let pY : p−1 (Y ) → Y
be the restriction of p to p−1 (Y ). Then (p−1 (Y ), pY , Y ) is a over Y and called the restriction of E
to Y. We denote (p−1 (Y ), pY , Y ) by E|Y .
Example 1.4. Let Y be a topological space and f : Y → X be a map. Suppose p : E → X is a
family over X. Equip the set Y × E with the product topology. Let f ∗ (E) be the subset of Y × E
consisting of points (y, e) so that f (y) = p(e), i.e.
f ∗ (E) = {(y, e) : f (y) = p(e)}.
Equip f ∗ (E) with the subspace topology induced from Y × E. Let f ∗ (p) : f ∗ (E) → Y be the
restriction of the projection πY : Y × E → Y to f ∗ (E). Then (f ∗ (E), f ∗ (p), Y ) is a family over Y
and called the induced family of p : E → X via f : Y → X.
Remark. If (E, p, X) is a family over X, we denote (E, p, X) by E when p and X are understood.
If ϕ is a morphism from (E, p, X) to (E 0 , p0 , X), we denote ϕ by ϕ : E → E 0 . If E and E 0 are
isomorphic families over X, we write E ∼
= E0.
One can check the following properties.
Proposition 1.1. Let f : Y → X and g : Z → Y be maps between topological spaces and p : E → X
be a family over X. Then we have
(1) g ∗ (f ∗ E) ∼
= (f ◦ g)∗ (E).
∗
∼
(2) f E = E|Y when Y is a subspace of X and f : Y → X is the inclusion map.
1For simplicity, a continuous family of vector spaces over X will be simply called a family over X in this note.