Vector space

• A vector space is a set $V$ with two operations of addition and scalar multiplication defined for its members, referred to as vectors.
  1. Vector addition maps any two vectors ${\bf x}, {\bf y}\in V$ to another vector ${\bf x}+{\bf y} \in V$ satisfying the following properties:
     • Commutativity: ${\bf x}+{\bf y}={\bf y}+{\bf x}$.
     • Associativity: ${\bf x}+({\bf y}+{\bf z})=({\bf x}+{\bf y})+{\bf z}$.
     • Existence of zero: there is a vector ${\bf0} \in V$ such that: ${\bf0}+{\bf x}={\bf x}+{\bf0}={\bf x}$.
     • Existence of inverse: for any vector ${\bf x} \in V$, there is another vector $-{\bf x}\in V$ such that ${\bf x}+(-{\bf x})={\bf0}$.
  2. Scalar multiplication maps a vector ${\bf x} \in V$ and a real or complex scalar $a\in C$ to another vector $a{\bf x} \in V$ with the following properties:
     • $a({\bf x}+{\bf y})=a{\bf x}+a{\bf y}$.
     • $(a+b){\bf x}=a{\bf x}+b{\bf x}$.
     • $ab{\bf x}=a(b{\bf x})$.
     • $1{\bf x}={\bf x}$.

• A metric space is a vector space $V$ in which the metric or distance $d({\bf x},{\bf y})$ between any two vector (or points) ${\bf x}$ and ${\bf y}$ is defined that satisfies:
  • $d({\bf x},{\bf y})$ iff ${\bf x}={\bf y}$
  • $d({\bf x},{\bf y})=d({\bf y},{\bf x})$
  • $d({\bf x},{\bf y})\le d({\bf x},{\bf z})+d({\bf z},{\bf y})$
  In an inner-product space in which the inner product $<{\bf x},{\bf y}>$ between any two vectors ${\bf x}$ and ${\bf y}$ is defined, the distance between the two points is $<({\bf x}-{\bf y}),({\bf x}-{\bf y})>$.