First of all I must apologize for the vague title and am open to suggestions.
This is not a Homework Assignment but something I once again encountered while reading a very compactly written paper.
$\tilde{\mathcal{C}}([0,1]^2,\mathbb{R}))$ is defined as follows for $f\in \tilde{\mathcal{C}}([0,1]^2,\mathbb{R}))$ the following holds:
- $f$ is continuous
- $f(x)=0 \quad \forall x\in [0,1]^2\setminus [0,1)^2$
I am looking for a countable family of continuous functions $\mathcal{F}:=(f_m)_{m\in\mathbb{N} }$ on $[0,1]^2$ so that the following requirements are satisfied
- $f\in \mathcal{F}\implies f(x)=0 \quad \forall x\in [0,1]^2\setminus [0,1)^2$
- the closure of the linear span of $(f_m)_{m\in\mathbb{N}}$ consists of all continuous Functions on $[0,1]^2$
According to the authors such a sequence is supposed to exist. But I do not know what the $f_m$ would like or why the family with the aforementioned properties exists. So I am looking for an explicit construction of the $f_m$ or a theorem which proofs the existence of such a sequence.
My first guess was $f_{n.m}(x_1,x_2):=x_1^nx_2^m$ for with Stone-Weierstrass it fullfills the second requirement. Unfortunately it does not meet the first one and I am somewhat at a loss.
Has anybody encountered a similar construction or knows of a theorem that might help?
As always thanks in advance :)