Consider the space CcR of compactly supported continuous fun

Consider the space C_c(R) of compactly supported continuous functions with distance function d_infinity. Is C_c(R) complete? Find all functions f epsilon C_c(R) for which one can find a sequence of polynomials p_n such that d_infinity (f, P_n) rightarrow 0.

Solution

(a) Consider the sequence of functions f_n(x) defined by

f_n (x) = 0 if |x| >=1

         =n (1-|x|) if 1-1/n <=|x|<1

          =1 if |x| < 1-1/n

Clearly all the f_n belong to C_c (R) and converge pointwise to the step function f which is 1 if |x|<1 and 0 otherwise.

So f is not continuous and this proved that C_c (R) is not complete.

(b) Such a function must have compact support as well as being approximable over entire R by polynomials. Looks like only the zero function has such a property