A Vector Space on the Set of Functions

We can define an operation on a set of functions of real numbers as follows.
  be a set of functions operating on a domain  
, which send each element of  
  into a codomain
, and let  
\[x \in D\]
. The set of all functions operating on  
  defines a vector space  
\[\mathbf{0}(x)=0 \in \mathbf{V}\]

2. For  
\[f,g \in \mathbf{V}, a,b \in \mathbb{R},af(x)+bg(x)=(af+bg)(x) \rightarrow af+bg \in \mathbf{V}\]