Given: S is a nonempty set, K is a field. Let C(S, K) denote the set of all functions ${f}\in\ C(S,K)$ such that ${f}(s) = 0 $ for all but a finite number of elements of S. Prove that C(S, K) is a vector space.
OK. I was thinking about using the simple additive axioms that define vector spaces. One of those is that there exist two elements such that $x$ (which is some vector) added to zero equals $x$, or $x + 0 = x$.
Let $g(s)$ be an arbitrary function. $f + g = g$ when $f(s) = 0$. In addition, if we assume $g(s)$ to be in the space $C(S,K)$ and $f + g = g$ then both vectors are in the space $C(S,K)$ and are closed under addition.
Am I on the right track here? I feel like there's another step I need to have.
Best Answer
Hints:
You first must define what the operations of sum and multiplication by a scalar are in that set $\,C(S,K)\,$ . These are pretty obvious, yet you must formally define them
Second, you must prove that under the definition above there's an element in $\,C(S,K)\,$ that serves as neutreal element of the sum.
Third, you must prove each element in $\,C(S,K)\,$ has an additive inverse.
Fourth, you must prove the corresponding axioms for multiplication by scalar.
Of the above, most is pretty simple and almost follows from the definitions.