For any two functions f and g in C([0,1],R), define
a. Prove that this defines a metric on C([0,1],R)
b. Prove that the following inequality relating this metric and the uniform metric:
c. Compare the concepts of convergence of a sequence of functions in this metric and in the uniform metric.
a. First, C([0,1],R) denotes that the function is continuous.
We need to prove that
i. if and only if f = g. If f = g, then
ii. d*(f,g) = d*(g,f) (symmetry): This is true since , thus
iii. Triangle Inequality:
Thus, d*(f,g) is a metric on C([0,1],R)
b.We need to prove that
I'm unsure where to start. I try to divide both sides by (b-a) which yields
Here, I use the Mean Value Theorem for integrals, since the function on [a,b] is continuous, then there exists a c such that
c. I'm not sure.
Thank you for your time.