Does the set (z^n ; n\in N) span L^2[0,1]?

[LikeMath]

Hey there,

Does the set (z^n ; n\in N) span L^2[0,1]?

Thanks in advance

 

[mathman]

http://staff.um.edu.mt/jmus1/hilbert.pdf

See page 17 of the above.

 

[LikeMath]

thank you very much

 

[Bacle2]

Maybe you could also use the following:

Polynomials are dense in C[a,b] (Weirstrass) ; Continuous functions ( in [a,b] , i.e., with compact support), are dense in simple functions, which are themselves dense in L²[a,b].

 

[micromass]

Maybe you could also use the following:

Polynomials are dense in C[a,b] (Weirstrass)

If you're going to use Weierstrass, then you have to know that this is for the $\| ~\|_\infty$ - norm. The theorem itself doesn't say anything for the $\|~\|_2$-norm.
Of course, on a compact interval, we have $\|~\|_2 \leq C \|~\|_\infty$ for some C that I'm too lazy to calculate. So density in $\|~\|_\infty$ would imply density in the $\|~\|_2$ norm.

 

[Bacle2]

O.K, good point, I was (implicitly) assuming that result for compact intervals.