Norms that $C([0,1])$ to be an incomplete normed space.

Question

I searched all of norms that $C([0,1])$ to be incomplete normed space. But I found only $\|.\|_p$ (for every $1\leq p<\infty$).

Are you know another norm on $C([0,1])$ that $C([0,1])$ to be incomplete normed space?

Of course I saw once another incomplete norm on $C([0,1])$ except $\|.\|_p$ . unfortunately I do not remember correct form it. But it was like to $\|f\|=\sum \frac{|f(r_n)|}{\cdots}$ that $\{r_n\}$ is sequence of rational numbers in $[0,1]$.

Is there everyone that know this norm correctly and exactly?

Answer 1

Let $(a_n) \in \ell^1$ be a sequence such that $a_n > 0$ (you maybe had in mind $a_n = 2^{-n}$) for all $n \in \mathbb N$, let $(r_n)$ be an enumeration of $\mathbb Q \cap [0,1]$. Then $$ f \mapsto \sum_{n=0}^\infty \left|f(r_n)\right|\cdot a_n $$ is a norm on $C([0,1])$. The definiteness follows from the fact, that a continuous function, vanishing on $\mathbb Q \cap [0,1]$ must vanish on the whole of $[0,1]$.

Answer 2

Most likely, the norm you saw was something like $$\|f\|=\sum_{n=1}^\infty\frac{|f(r_n)|}{2^n}$$ – where $(r_n)$ is any dense sequence in $[0,1]$, for example, one denumerating all the rationals in $[0,1]$. The factor $2^n$ in the denominator is there just to assure that the series converges. You could just as well put $n^2$ there, for example.

It turns out that a sequence $(f_k)$ converges in this norm to a function $f$ if and only if $f_k(r_n)\to f(r_n)$ for all $n$. (Try proving it yourself.) From this, you can probably find Cauchy sequences that don't converge; for a simple example, take a sequence of continuous functions converging pointwise to the characteristic function of $[0,\frac12]$.

Answer 3

There are several generalizations of $L^p$ spaces:

Weighted $L^p$ spaces. A function $w\colon[0,1]\to[0,\infty)$ is called a wight. $$ \|f\|_{L^p(w)}=\Bigl(\int_0^1|f(x)|^pw(x)\,dx\Bigr)^{1/p},\quad 1\le p<\infty. $$

Orlicz's spaces

Lorentz spaces