Chapter 4.1, Problem 20E

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].

1. 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.)
2. b. Show that {f in C[a, b] : f(a) = f(b)} is a subspace of C[a, b].

For fixed positive integers m and n, the set M_m×n of all m x n matrices is a vector space, under the usual operations of addition of matrices and multiplication by real scalars.