Is this ball in C([0,1]) compact?

1. Apr 5, 2009

andrassy

1. The problem statement, all variables and given/known dataLet C([0,1]) be the metric space of continuous functions on the interval [0,1] with distance = max of x over [0,1] of |f(x)-g(x)|. Is the ball of radius 1 centered around f(x) = 0 compact?

3. The attempt at a solutionI originally thought it was but now I believe that it is not compact. I'm not sure how to prove it though. I know I can use either sequential convergence or show that it isn't totally bounded, but this is where I get stuck. I know the ball has all continuous functions s.t. |f(x)| < 1. How can I go about showing it isnt sequentially compact or that it isnt totally bounded? Anyone can put me in the right direction?

2. Apr 6, 2009

Are you sure it's the open ball, with |f| < 1? In that case, it's easily not compact (it's not closed).

The closed ball would be harder, but look up Ascoli's theorem. Is the ball equicontinuous?

Last edited: Apr 6, 2009
3. Apr 6, 2009

maze

This is a great question!

In a metric space, compactness is equivalent to sequential compactness, so we only need to determine if every sequence of points in the ball has a convergent subsequence.

In a n-dimensional vector space, you could construct a finite sequence f1, f2, ..., fn with each fi far apart from all the others by placing each fi on a different perpendicular axis. eg: in 3d f1=(1,0,0), f2=(0,1,0), f3=(0,0,1). Leting n go to infinity, in an infinite dimensional vector space theoretically you should be able to create an infinite sequence with no convergent subsequence using a similar argument.

The problem arises that C([0,1]) does not have an inner product, so you can't construct a "orthogonal" set of vectors. However, it is a normed vector space so you can effectively do the equivalent by applying Reisz's lemma to construct a sequence of unit vectors, each one of which is at least a fixed distance from the subspace spanned by all the previous vectors.

Last edited: Apr 6, 2009