"Let V denote the set of all differentiable real-valued functions defined on the real line."
Does this automatically mean that this vector space is over the field of reals?
Why or why not?
I ask because I need to prove this is a vector space. But, if I pick some element a from F (the field), then the scalar multiplication of a and an element of V is only real-valued if a is a real. This would make this scalar multiplication not an element of V, making it not a vector space, if F were a field that contained non-real elements. So, I must assume that this V is over the field of reals in order for it to prove it is a vector space, but why am I warranted to make that claim?
Sorry if this is a dumb question, I am just starting LA independently.