Let V be the set of all real-valued continuous functions. If f and g are in V, define f+g by (f+g)(t) =?

Let V be the set of all real-valued continuous functions. If f and g are in V, define f+g by (f+g)(t) = f(t)+g(t). If f is in V, define c.f by (c.f)(t) = cf(t). Prove that V is a vector space, which is denoted by C(-infinity, infinity).





By definition, a vector space is a set V of elements on which we have two operations '+' and '.' defined with the following properties:





a) If u and v are any elements in V, then u+v is in V. (We say that V is closed under the operation '+')





b) If u is any element in V and c is any real number, then c.u is in V.( i.e., V is closed under the operation '.')





Here, the operation '+' is vector addition and the operation '.' is scalar multiplication.

Let V be the set of all real-valued continuous functions. If f and g are in V, define f+g by (f+g)(t) =?
V is not only a vector space, it is, in fact, a linear vector space!





To verify the requirements,





a) prove that if f,g are real, continuous, the f+g is also real, continuous.





The reals are closed under addition, and continuity can be proven from epsilon-delta arguments.





b) if u is real, continuous, and c a scalar in R, you only need to prove that cu is real, and continuous. R is closed under multiplication, and the cont of cu, again, from definition, some epsilon-delta argument.