Definition VS Vector Space

Suppose that $V$ is a set upon which we have defined two operations: (1) vector addition, which combines two elements of $V$ and is denoted by “+”, and (2) scalar multiplication, which combines a complex number with an element of $V$ and is denoted by juxtaposition. Then $V$, along with the two operations, is a vector space over $\complexes$ if the following ten properties hold.

• AC Additive Closure
  If $\vect{u},\,\vect{v}\in V$, then $\vect{u}+\vect{v}\in V$.
  
  
• SC Scalar Closure
  If $\alpha\in\complex{\null}$ and $\vect{u}\in V$, then $\alpha\vect{u}\in V$.
  
  
• C Commutativity
  If $\vect{u},\,\vect{v}\in V$, then $\vect{u}+\vect{v}=\vect{v}+\vect{u}$.
  
  
• AA Additive Associativity
  If $\vect{u},\,\vect{v},\,\vect{w}\in V$, then $\vect{u}+\left(\vect{v}+\vect{w}\right)=\left(\vect{u}+\vect{v}\right)+\vect{w}$.
  
  
• Z Zero Vector
  There is a vector, $\zerovector$, called the zero vector, such that $\vect{u}+\zerovector=\vect{u}$ for all $\vect{u}\in V$.
  
  
• AI Additive Inverses
  If $\vect{u}\in V$, then there exists a vector $\vect{-u}\in V$ so that $\vect{u}+ (\vect{-u})=\zerovector$.
  
  
• SMA Scalar Multiplication Associativity
  If $\alpha,\,\beta\in\complex{\null}$ and $\vect{u}\in V$, then $\alpha(\beta\vect{u})=(\alpha\beta)\vect{u}$.
  
  
• DVA Distributivity across Vector Addition
  If $\alpha\in\complex{\null}$ and $\vect{u},\,\vect{v}\in V$, then $\alpha(\vect{u}+\vect{v})=\alpha\vect{u}+\alpha\vect{v}$.
  
  
• DSA Distributivity across Scalar Addition
  If $\alpha,\,\beta\in\complex{\null}$ and $\vect{u}\in V$, then $(\alpha+\beta)\vect{u}=\alpha\vect{u}+\beta\vect{u}$.
  
  
• O One
  If $\vect{u}\in V$, then $1\vect{u}=\vect{u}$.
  
  

The objects in $V$ are called vectors, no matter what else they might really be, simply by virtue of being elements of a vector space.