Product of Vector Spaces

Let ${(V_{i}, +, \cdot)}_{i = 0}^{n}$ be a collection of vector spaces over the field $F$ . Then the product of vector spaces is:

\prod_{i = 0}^{n} V_{i} = {(v_{i})_{i = 0}^{n} ∣ \forall 0 \leq i \leq n [v_{i} \in V_{i}]}

with the following operations, let $(v_{i})_{i = 0}^{n}, (w_{i})_{i = 0}^{n} \in \prod V_{i}$ , and $λ \in F$ :

(v_{i})_{i = 0}^{n} + (w_{i})_{i = 0}^{n} = (v_{i} + w_{i})_{i = 0}^{n} λ (v_{i})_{i = 0}^{n} = (λ v_{i})_{i = 0}^{n}

Then $\prod V_{i}$ is a vector space over the field $F$ .

Given that every $V_{i}$ is finite dimensional vector spaces, then:

\dim (\prod_{i = 0}^{n} V_{i}) = \sum_{i = 0}^{n} \dim (V_{i})

Let ${U_{i}}_{i = 0}^{n}$ be a collection of subspaces of $V$ . Defining a linear map $Γ : \prod U_{i} \to \sum U_{i}$ , such that:

Γ (u_{i})_{i = 0}^{n} = \sum_{i = 0}^{n} u_{i}

$Γ$ is injective iff $\sum U_{i}$ is a direct sum.