Prove that the algebra generated is dense

[fabiancillo]

Hello I have problems with this exercise

Prove that the algebra generated by the set $S = \{ 1,x^2 \}$ is dense in $C [0, 1]$. It is $S$ dense in $C [-1; 1]$

I am thinking to apply Stone-weierstrass theorem but I don't know how to use it properly.

Thanks

 

[Euge]

Hi cristianoceli,

That is a good idea. Since $[0,1]$ is compact Hausdorff and $S$ is a subalgebra of $C[0,1]$, it suffices to show that $S$ contains a nonzero constant function and separates points. By definition of $S$, the nonzero constant function $1$ belongs to $S$. Furthermore, if $a,b\in [0,1]$ with $a\neq b$, the function $p(x) = x^2$ is an element in $S$ such that $p(a) \neq p(b)$. Thus $S$ separates points in $C[0,1]$.

On the other hand, $S$ is not dense in $C[-1,1]$, since $S$ contains only even functions and the function $f(t) = t$ is odd.

 

[fabiancillo]

Thansk Good idea