1. The problem statement, all variables and given/known data For the real-valued function All f with 2f(0)=f(1) with domain [0, 1], determine whether the given set is a real linear space, if addition and multiplication by real scalars are defined in the usual way. For those that are not, tell which axioms fail to hold. 2. Relevant equations Supposed to use the Closure axioms, axioms for additions, and axioms for multiplication for a proof. 3. The attempt at a solution To me these axioms are trivial and it is obvious that the function is a real linear space; however, the question wants me to prove them. Here are my attempts: In the following situations, let f, g, h be functions that satisfy the condition 2f(0) = f(1). Axiom 1 (Closure under addition):Then 2f(0) + 2g(0) = f(1) + g(1). Axiom 2 (Closure under multiplication): Let a be a real number. Therefore af(0) = af(1). Axiom 3 (Commutative law): 2f(0) + 2g(0) = f(1) + g(1) Axiom 4 (Assosciative law): 2f(0) +) 2g(0) + 2h(0)) = (f(1) + g(1)) +h(1) My other arguments follow in this way. Is this enough of a proof for each statement? Is there a better way to work through this?