1.2 Vector spaces

Metric spaces impose no requirements on the structure of the set $M$ . We will now consider more structured $M$ , beginning by generalizing the familiar concept of a vector.

Definition 1

1. vector addition: $+:V\times V\to V$
2. scalar multiplication: $·:K\times V\to V$

We say that $V$ is a vector space (or linear space ) over $K$ if

• VS1: $V$ forms a group under addition, i.e.,
  • $(x+y)+z=x+(y+z)$ (associativity)
  • $x+y=y+x$ (commutativity)
  • $\exists 0\in V$ such that $\forall x\in V$ , $x+0=0+x=x$
  • $\forall x\in V$ , $\exists y$ such that $x+y=0$
• VS2: For any $\alpha ,\beta \in K$ and $x,y\in V$
  • $\alpha \left(\beta x\right)=\left(\alpha \beta \right)x$ (compatibility)
  • $(\alpha +\beta )(x+y)=\alpha x+\alpha y+\beta x+\beta y$ (distributivity)
  • $\exists 1\in K$ such that $1x=x$