Vector Space

A non empty set $V$ over a field $F$ satisfying with 2 binary operations, commonly denoted as $+$ and $\times$, satisfying:

$$\text{Properties:}\{\begin{array}{rl}u+(v+w)=(u+v)+w& \text{Addition of vectors is associative}\\ u+v=v+u& \text{Addition is commutative}\\ \exists 0\in V:v+0=v\forall v\in V& \text{There is an additive identity vector 0∈V}\\ \forall v\in V\exists (-v):(-v)+v=0& \text{All vectors have an additive inverse}\\ a(bv)=(ab)v& \text{Scalar multiplications is associative with field multiplication}\\ 1v=v& \text{There is a multiplicative identity scalar 1∈F}\\ a(u+v)=au+av& \text{Scalar multiplication is distributive over vector addition}\\ (a+b)V=av+bv& \text{Scalar multiplication is distributive over field addition}\end{array}$$

In the above,

• $u,v,w\in V$ are vectors
• $a,b\in F$ are scalars

Examples

• Banach Space
• Hilbert Space
• Phase Space