Does the set (z^n ; n\in N) span L^2[0,1]?
Thanks in advance
See page 17 of the above.
thank you very much
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].
If you're going to use Weierstrass, then you have to know that this is for the [itex]\| ~\|_\infty[/itex] - norm. The theorem itself doesn't say anything for the [itex]\|~\|_2[/itex]-norm.
Of course, on a compact interval, we have [itex]\|~\|_2 \leq C \|~\|_\infty[/itex] for some C that I'm too lazy to calculate. So density in [itex]\|~\|_\infty[/itex] would imply density in the [itex]\|~\|_2[/itex] norm.
O.K, good point, I was (implicitly) assuming that result for compact intervals.
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