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

  • Context: Graduate 
  • Thread starter Thread starter LikeMath
  • Start date Start date
  • Tags Tags
    Set Span
Click For Summary

Discussion Overview

The discussion centers around whether the set of functions of the form (z^n ; n∈N) spans the space L^2[0,1]. The scope includes theoretical considerations related to function spaces and density arguments in analysis.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant asks if the set (z^n ; n∈N) spans L^2[0,1].
  • Another participant references a document that may contain relevant information regarding the question.
  • A different participant suggests that polynomials are dense in C[a,b] according to the Weierstrass theorem, which may relate to the original question.
  • It is noted that while Weierstrass's theorem applies to the ||~||∞ norm, it does not directly address the ||~||2 norm, raising a point about the implications of density in different norms.
  • One participant acknowledges an implicit assumption regarding the result for compact intervals, indicating a potential oversight in their reasoning.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of the Weierstrass theorem to the context of L^2[0,1], and there is no consensus on whether the set (z^n ; n∈N) spans L^2[0,1].

Contextual Notes

The discussion includes assumptions about the relationship between different norms and the implications of density in function spaces, which remain unresolved.

LikeMath
Messages
62
Reaction score
0
Hey there,

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

Thanks in advance
 
Physics news on Phys.org
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].
 
Bacle2 said:
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.
 
Last edited:
O.K, good point, I was (implicitly) assuming that result for compact intervals.
 

Similar threads

Undergrad Create a surjective function from [0,1]^n→S^n
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad (0,1) is uncountable using binary expansions?
  • · Replies 15 ·
Replies
15
Views
4K
Undergrad Convergence not defined by any metric
  • · Replies 17 ·
Replies
17
Views
2K
Graduate Understanding Cantor set C in Ternary form with 1/n factor in front C
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Show that set is an orthonormal set in PC(0,l)
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Shauder basis for Hilbert spaces
  • · Replies 0 ·
Replies
0
Views
710
Undergrad Why is the dual of Z^n again Z^n ?
  • · Replies 13 ·
Replies
13
Views
2K
Undergrad I need a mapping from the unit Hypercube [0,1]^n to a given simplex
  • · Replies 17 ·
Replies
17
Views
5K
Undergrad Functions.wolfram.com accepted and published my formula for Gamma Fn
  • · Replies 11 ·
Replies
11
Views
2K
Graduate Laurent series for algebraic functions
  • · Replies 9 ·
Replies
9
Views
3K