Let V be the Vector space of all realvalued functions define

Let V be the Vector space of all real-valued functions defined and continuous on (-infinity, infinity). For each of the following sets S, determine whether S is a subspace of V. If the set does form a subspace, show that closure under addition and scalar multiplication hold. If the set does not form a subspace, explain why not.

(a) S is the set of all function f satisfying f(-x) = f(x)

(b) S is the set of all function f satisfying f(0) = 1

(c) S is the set of all function f satisfying f(-x) = -f(x)

(d) S is the set of all function f satisfying lim (x --> infinity) f(x) = 0 and lim (x--> - infinity)f(x) = 0.

Solution