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3. Let f : S → S ′ be any map of sets. Applying in the diagram
S
ι S
f
S ′ ι S ′
V (S)
V (f)
V (S ′ )
the defining property of the free vector space V (S) to the map ι S ′ ◦ f of sets, we find a
unique K-linear map V (f) : V (S) → V (S ′ ).
One checks that this defines a functor V : Set → vect(K). For any K-vector space W and
any set S, we have a bijection of morphism spaces:
Hom K (V (S), W ) ∼ = Hom Set (S, U(W ))
that is compatible with morphism of sets and K-linear maps. One says that the functor
V is a left adjoint to the forgetful functor U.
A.2 Tensor products of vector spaces
We summarize some facts about tensor products of vector spaces over a field K.
Definition A.2.1
Let K be a field and let V, W and X be K-vector spaces. A K-bilinear map is a map
α : V × W → X
that is K-linear in both arguments, i.e. α(λv + λ ′ v ′ , w) = λα(v, w) + λ ′ α(v ′ , w) and α(v, λw +
λ ′ w ′ ) = λα(v, w) + λ ′ α(v, w ′ ) for all λ, λ ′ ∈ K and v, v ′ ∈ V , w, w ′ ∈ W .
Given any K-linear map ϕ : X → X ′ , the map ϕ ◦ α : V × W → X ′ is K-bilinear as well.
This raises the question of whether for two given K-vector spaces V, W , there is a “universal”
K-vector space with a universal bilinear map such that all bilinear maps out of V × W can be
described in terms of linear maps out of this vector space.
Definition A.2.2
The tensor product of two K-vector spaces V, W is a pair, consisting of a K-vector space V ⊗W
and a bilinear map
κ : V × W → V ⊗ W
(v, w) ↦→ v ⊗ w
with the following universal property: for any K-bilinear map
α : V × W → U
there exists a unique linear map ˜α : V ⊗ W → U such that
α = ˜α ◦ κ .
As a diagram:
V × W
U
κ
∃!˜α
V ⊗ W
α
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