## 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 \[V\]

aince1.

\[\mathbf{0}(x)=0 \in \mathbf{V}\]

2. For

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