Prove that the algebra generated is dense

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S = \{1, x^2\}$ is dense in $C[0,1]$, we can apply the Stone-Weierstrass theorem. This is because $[0,1]$ is compact Hausdorff and $S$ is a subalgebra of $C[0,1]$. To use the theorem, we need to show that $S$ contains a nonzero constant function and separates points. Both of these conditions can be met since $S$ contains the function $1$ and also the function $x^2$ which can separate points. However, $S$ is not dense in $C[-1,1]$ because it only
  • #1
fabiancillo
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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
 
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  • #2
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.
 
  • #3
Thansk Good idea
 

1. What does it mean for an algebra to be dense?

For an algebra to be dense, it means that the elements in the algebra can approximate any real number with arbitrary precision. In other words, the algebra contains a "dense" set of elements that can get arbitrarily close to any real number.

2. How do you prove that an algebra is dense?

To prove that an algebra is dense, you must show that for any real number x, there exists a sequence of elements in the algebra that converge to x. This can be done by using the definition of a dense set and showing that the algebra satisfies this property.

3. Can an algebra be dense in a specific interval?

Yes, an algebra can be dense in a specific interval. This means that the elements in the algebra can approximate any real number within that interval with arbitrary precision. However, the algebra may not be dense in other intervals or in the entire real number line.

4. Is it possible for an algebra to be dense in one set of numbers but not in another?

Yes, it is possible for an algebra to be dense in one set of numbers but not in another. For example, an algebra may be dense in the set of rational numbers but not in the set of irrational numbers. This is because the algebra may only contain elements that can approximate rational numbers but not irrational numbers.

5. How is the density of an algebra related to its completeness?

The density of an algebra is closely related to its completeness. A complete algebra is one in which all Cauchy sequences converge to an element in the algebra. This means that the algebra is dense and can approximate any real number. However, an algebra can be dense without being complete. In this case, there may be Cauchy sequences that do not converge to an element in the algebra.

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