Question: The set of all continuous real-valued functions defined on a closed interval [a, b] in R is denoted by C [a, b]. This set is a subspace of the vector space of all real-valued functions defined on [a, b].
a. What facts about continuous functions should be proved in order to demonstrate that C [a, b] is indeed a subspace as claimed? (These facts are usually discussed in a calculus class.)
b. Show that {f in C [a, b] : f (a) = f(b)} is a subspace of C [a, b].