(C[0,1],|| ||2) is a complete metric space

[catcherintherye]

Homework Statement

I am required to show that (C[0,1], || ||2) is a complete metric space, or to disprove that it is

Homework Equations

C[0,1] is the set of continuous functions on the bounded interval 0,1

The Attempt at a Solution

I am immediately confused as I am told in my notes that if every cauchy sequence from X converges then (X,d) is a complete metric space, but I find it hard to see that there clould exist a divergent cauch sequencey since cauchy sequences are all convergent right?

 

[StatusX]

Cauchy sequences converge, but not necessarily to something in the space (every metric space has a completion in which every cauchy sequence converges, and which the original metric space is a dense subset of). Can you find a cauchy sequence of continuous functions whose limit isn't continuous?

 

[catcherintherye]

would such an example be the sequence of spike functions? i.e the sequence of functions gn on [0,1] for which gn(x) is n2x when x is in [0,1/n], n(2-nx) when x is in [1/n,2/n] and zero elsewhere. The pointwise limit of gn(x) is always zero (x=0 included, since gn(0)=0 for any n). Yet, the integral of gn is always equal to 1,

 

[HallsofIvy]

Homework Statement

I am required to show that (C[0,1], || ||2) is a complete metric space, or to disprove that it is

Homework Equations

C[0,1] is the set of continuous functions on the bounded interval 0,1

The Attempt at a Solution

I am immediately confused as I am told in my notes that if every cauchy sequence from X converges then (X,d) is a complete metric space, but I find it hard to see that there clould exist a divergent cauch sequencey since cauchy sequences are all convergent right?

Cauchy sequences in a complete metric space converge- that's the definition of "complete"!

Perhaps you are thinking of the fact that Cauchy sequences of real numbers converge- the real numbers with the "usual" metric is complete.

Here, your metric space is functions continuous on [0,1] so your "Cauchy sequences" are sequences of continuous functions. You need to show that Cauchy sequence of continuous functions converge. What is the precise definition of || ||2 ?

 

[catcherintherye]

|| ||2 is the 2-norm so ||f||2 =the square root of the integral from 0 to 1 of [f(x)^2]dx

 

[HallsofIvy]

That finally dawned on me! I was thinking of the infinite case where continuous function may not be square integrable.