Is the following space $(C_p(\infty))$ Banach?

Question

I have proven (i),(ii) but I suspect (iii) is not Banach. But I can't seem to come up with any counter examples any ideas?

Answer 1

Let $f_n(x)=0$ for $x \leq \frac 1 2$, $nx-\frac n 2$ for $\frac 1 2 \leq x \leq \frac 1 2 +\frac 1 n$ and $1$ for $ \frac 1 2 +\frac 1 n\leq x \leq 1$, $2-x$ for $1 \leq x \leq 2$ and $0$ for $x \geq 2$. This sequence is Cauchy. If $f_n \to g$ in this space the there is a subsequence of $\{f_n\}$ converging almost everywhere to $f$. But the pointwise limit of $f_n$ is the characteristic function of $[\frac 1 2 ,1]$. Now show that this characteristic function cannot be equal almost eveywhere to a continuous function.

Answer 2

Hint: You want to find a Cauchy-sequence that does not converge. Hence, you may want to find a sequence that "wants to converge", but cannot because its limit is not continuous and hence not in $C_p(\mathbb R)$. Certain classical examples of sequences that converge pointwise but not uniformly will do the trick.