In my homework I am asked to show that:
1) is a norm on [a,b] where is defined by = (the second is meant to be 'f dash' i.e. the derivative of f )
2) [a,b] is complete in this norm.
My attempt so far:
1) Fairly easy, just show that the norm satisfies the three conditions in the definition of a norm.
2) This is what I'm stuck with.
is complete if :
Given a Cauchy sequence in
such that the limit of the sequence is
Choose a sequence of functions in
converging uniformly to
such that . Then by definition is complete.
PROBLEM: Is there any reason that there is such a sequence? Obviously I would have to show somehow why such a sequence exists ... and unfortunately I don't know how to do this because in my notes there isn't any examples to show why, for example, any finite-dimensional normed vector space is complete over .
N.B. Sorry about the format, it's the first time I've used latex!