Let $S$ be a [[set]] and $K$ a [[field]]. By a [[function]] of $S$ into $K$ we shall mean an association which to each [[Element of a Set|element]] of $S$ associates a unique element of $K$. Thus if $f$ is a function of $S$ into $K$, we express this by the symbols $ f: S \rightarrow K . $ We also say that $f$ is a $K$-valued function. Let $V$ be the set of all functions of $S$ into $K$. If $f, g$ are two such functions, then we can form their sum $f+g$. It is the function whose value at an element $x$ of $S$ is $f(x)+g(x)$. We write $ (f+g)(x)=f(x)+g(x) $ If $c \in K$, then we define $c f$ to be a scaled output of $f$ $ (c f)(x)=c f(x) $