Incomplete space

Question

I must show that $C_0(\mathbb{R})$ space of all continuous real functions $f:\mathbb{R} \longrightarrow \mathbb{R}$ with compact support is not a complete space endowed by norm $\|f\|=\sup\limits_{t \in \mathbb{R}}|f(t)|$.

Thank for any help.

score 2 · Answer 1 · answered May 21 '13 at 07:02

2

Hint: find a sequence whose limit does not have compact support.

answered May 21 '13 at 07:02

Robert Israel

448,999

score 1 · Answer 2 · answered May 21 '13 at 07:36

1

Hint: You can use indicator functions.

answered May 21 '13 at 07:36

Seirios

33,157

score 0 · Answer 3 · answered May 21 '13 at 08:16

0

You could also use Baire's theorem to obtain $n\in \mathbb N$ such that every element of $C_0(\mathbb R)$ would have support in $[-n,n]$.

answered May 21 '13 at 08:16

Jochen

12,254

Incomplete space

3 Answers3