how to prove that the set "all f with f(0) = f(1)", function has domain containing 0 and 1; is a linear space?
Do you know what a linear space is?
The standard way of proving "X is Y" is to use the definition of X to show that it also satisfies the definition of Y.
Here, X is the all functions, f, such that f(0)= f(1). Now, you need to show that that set satisfies the definition of "linear space"- that's just what Plato is asking.