Definition:Direct Product of Vector Spaces

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Definition

Let $K$ be a field.

Let $V, W$ be $K$-vector spaces.

The direct product of $V$ and $W$ is their module direct product.


Finite Case

Let $K$ be a field.

Let $V_1, V_2, \ldots, V_n$ be $K$-vector spaces.

Let:

$\displaystyle V = \prod_{k \mathop = 1}^n V_k$

be their cartesian product.

Let:

$+$ be the operation induced on $V$ by the operations $+_1, +_2, \ldots, +_n$ on $V_1, V_2, \ldots, V_n$
$\circ$ be defined as $\lambda \circ \tuple {x_1, x_2, \ldots, x_n} := \tuple {\lambda \circ x_1, \lambda \circ x_2, \ldots, \lambda \circ x_n}$


$\struct {V, +, \circ}_K$ is called the direct product of $V_1, \ldots, V_n$.


General Case

Let $K$ be a field.

Let $\family {V_i, +_i, \circ_i}_{i \mathop \in I}$ be a family of $K$-vector spaces.


The (external) direct product of $\family {V_i, +_i, \circ_i}_{i \mathop \in I}$ is their module direct product.


Also see