## A Vector Space on the Set of Functions

We can define an operation on a set of functions of real numbers as follows.
Let
$F$
be a set of functions operating on a domain
$D$
, which send each element of
$D$
into a codomain
$C$
, and let
$x \in D$
. The set of all functions operating on
$x$
defines a vector space
$V$
aince
1.
$\mathbf{0}(x)=0 \in \mathbf{V}$

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