- #1
jgthb
- 8
- 0
Hi everyone
Today during problem session we had this seemingly simple exercise, but I just can't crack it:
We should give an example of an [tex]x \in \ell^2[/tex] with strict inequality in the Bessel inequality (that is an x for which [tex]\sum_{k=1}^\infty |<x,x_k>|^2 < ||x||^2[/tex], where [tex](x_k)[/tex] is an orthonormal basis). I have tried a few things, e.g. defining [tex]x_k[/tex] in the following way:
[tex]x_1 = (\frac{1}{\sqrt(2)},\frac{1}{\sqrt(2)},0,\ldots)[/tex]
[tex]x_2 = (\frac{-1}{\sqrt(2)},\frac{1}{\sqrt(2)},0,\ldots)[/tex]
[tex]x_k = (0,\ldots,0,1,0,\ldots)[/tex], for [tex]k \geq 3[/tex]
and defining x as [tex]x = (1,1,0,0,\ldots)[/tex],
but that doesn't seem to work. Does anyone have a better idea?
Today during problem session we had this seemingly simple exercise, but I just can't crack it:
We should give an example of an [tex]x \in \ell^2[/tex] with strict inequality in the Bessel inequality (that is an x for which [tex]\sum_{k=1}^\infty |<x,x_k>|^2 < ||x||^2[/tex], where [tex](x_k)[/tex] is an orthonormal basis). I have tried a few things, e.g. defining [tex]x_k[/tex] in the following way:
[tex]x_1 = (\frac{1}{\sqrt(2)},\frac{1}{\sqrt(2)},0,\ldots)[/tex]
[tex]x_2 = (\frac{-1}{\sqrt(2)},\frac{1}{\sqrt(2)},0,\ldots)[/tex]
[tex]x_k = (0,\ldots,0,1,0,\ldots)[/tex], for [tex]k \geq 3[/tex]
and defining x as [tex]x = (1,1,0,0,\ldots)[/tex],
but that doesn't seem to work. Does anyone have a better idea?