# ベクトル空間の定義

## Definition Vector Space

Let $V$ is a Set.
operations are defined as
$f:V \times V \rightarrow V$
$g : \mathbb{R} \times V \rightarrow V$
(* $\mathbb{R}$ はより一般に"体"に置き換えることができる.) $f(x, y)$ is expressed by $x + y$
$g(x, y)$ is expressed by $xy$
If $V$ fills next conditions, then
$V$ is a Vector Space. $a, b, c \in V$
$r, s \in \mathbb{R}$
(1) $(a + b) + c = a + (b + c)$
(2) $\exists 0 \in V, \forall v \in V \quad s.t. \quad 0 + v = 0$
(3) $\forall v \in V, \exists -v \in V, \quad s.t. \quad v + (-v) = 0$
(4) $a + b = b + a$
(5) $r (a + b) = ra + rb$
(6) $(r + s)a = ra + sa$
(7) $(rs)a = r(sa)$
(8) $1a = a$ (1 は $\mathbb{R}$ の単位元)