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 kn with the topology defined by the Euclidean norm kak =p|a1|2+ · · · + |an|2
for any a = (a1, · · · , an) ∈ kn. This topology is called the Euclidean topology on kn. One can see that kn 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 : V −−−−→ W × W(f,g) −−−−→ Wα
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 Exa topological vector space isomorphic to knfor 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 Exover x is the set Ex= {(x, v) ∈ E : v ∈ V }. For (x, v) and (x, w) in Exand a ∈ k, we define
(x, v) +x(x, w) = (x, v + w), a ·x(x, v) = (x, av).
1
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 (E0, p0, X) is a continuous map ϕ : E → E0 such that p0= ϕ◦p. The set of all morphisms from (E, p, X) to (E0, p0, X) is denoted by Hom(E, E0).
An isomorphism ϕ from (E, p, X) to (E0, p0, X) is a morphism such that ϕ : E → E0 is a bijection whose inverse ϕ−1 : E0→ E determines a morphism from (E0, p0, X0) 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 (E0, p0, X), we denote ϕ by ϕ : E → E0. If E and E0 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.
