I am a beginning math student in linear algebra. We are going through Vector spaces and subspaces. The question is: Which of the following sets are vector spaces? Give a proof or a counterexample. The set of even functions: $\{f:\Bbb R \to\Bbb R \mid f(-x)=f(x)\text{ for all }x\in\Bbb R\}\;,$ with the usual scalar multiplication and addition of functions.
Is it ok to just write down the 10 axioms that make something a vector space, and then change the variable names to make it more my own? How does writing the axioms down make the proof true?