Unit ball in $C[0,1]$ not sequentially compact

sequences-and-series compactness convergence-divergence functional-analysis uniform-convergence

This question is taken from Saxe K -Beginning Functional Analysis.

Show that the closed unit ball in $C[0,1]$ is not compact by proving that it is not sequentially compact.

(It's assumed that we are using the uniform norm).

I've been working on this for ages but I could not come up with any sequence $\{f_n\}$ in the unit ball such that there exists $N\in \mathbb{N}$ such that for all $m,n\geq N$ we have that $d(f_n,f_m)>c$. Should be a nice example of this, please help me!


Solution 1:

Consider $f_n(t)=t^n$, $0\le t\le 1$. Then $\{f_n\} \subset \overline{B(0,1)}$ (closed unit ball), but no subsequence of $\{f_n\}$ converges in $C[0,1]$ (with the sup norm).

Solution 2:

Hint: each subsequence should converge uniformly to the pointwise limit, which is not continuous.

So take any bounded sequence in $C[0,1]$ which converges pointwise to a non-continuous function.

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