Prove that $x_i$ converges weakly to $y$.
$begingroup$
Let $H$ be a Hilbert space. Let ${x_i}$ be a sequence in $H$ with the following two properties:
$|x_i|=1$ for all $i$
- There is a fixed number $cgeq0$ such that $(x_i,x_j)=c$ whenever $ineq j$.
Define
$$
y_n:=frac{1}{n}sum_{i=1}^nx_i.
$$
Prove that there is a $yin H$ such that $y_n$ converges strongly to $y$ and $x_i$ converges weakly to $y$.
I can prove that $y_n$ converges strongly to $y$ by proving it is a Cauchy sequence. But I don't know how to prove $x_i$ converges to $y$ weakly.
functional-analysis
asked Dec 7 '18 at 2:38
whereamIwhereamI
320115
$endgroup$
add a comment |
$begingroup$
Let $H$ be a Hilbert space. Let ${x_i}$ be a sequence in $H$ with the following two properties:
$|x_i|=1$ for all $i$
- There is a fixed number $cgeq0$ such that $(x_i,x_j)=c$ whenever $ineq j$.
Define
$$
y_n:=frac{1}{n}sum_{i=1}^nx_i.
$$
Prove that there is a $yin H$ such that $y_n$ converges strongly to $y$ and $x_i$ converges weakly to $y$.
I can prove that $y_n$ converges strongly to $y$ by proving it is a Cauchy sequence. But I don't know how to prove $x_i$ converges to $y$ weakly.
functional-analysis
asked Dec 7 '18 at 2:38
whereamIwhereamI
320115
$endgroup$
add a comment |
$begingroup$
Let $H$ be a Hilbert space. Let ${x_i}$ be a sequence in $H$ with the following two properties:
$|x_i|=1$ for all $i$
- There is a fixed number $cgeq0$ such that $(x_i,x_j)=c$ whenever $ineq j$.
Define
$$
y_n:=frac{1}{n}sum_{i=1}^nx_i.
$$
Prove that there is a $yin H$ such that $y_n$ converges strongly to $y$ and $x_i$ converges weakly to $y$.
I can prove that $y_n$ converges strongly to $y$ by proving it is a Cauchy sequence. But I don't know how to prove $x_i$ converges to $y$ weakly.
functional-analysis
asked Dec 7 '18 at 2:38
whereamIwhereamI
320115
$endgroup$
Let $H$ be a Hilbert space. Let ${x_i}$ be a sequence in $H$ with the following two properties:
$|x_i|=1$ for all $i$
- There is a fixed number $cgeq0$ such that $(x_i,x_j)=c$ whenever $ineq j$.
Define
$$
y_n:=frac{1}{n}sum_{i=1}^nx_i.
$$
Prove that there is a $yin H$ such that $y_n$ converges strongly to $y$ and $x_i$ converges weakly to $y$.
I can prove that $y_n$ converges strongly to $y$ by proving it is a Cauchy sequence. But I don't know how to prove $x_i$ converges to $y$ weakly.
functional-analysis
functional-analysis
asked Dec 7 '18 at 2:38
whereamIwhereamI
320115
asked Dec 7 '18 at 2:38
whereamIwhereamI
320115
asked Dec 7 '18 at 2:38
whereamIwhereamI
320115
asked Dec 7 '18 at 2:38
whereamIwhereamI
320115
asked Dec 7 '18 at 2:38
whereamIwhereamI
320115
320115
add a comment |
add a comment |
1 Answer
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$begingroup$
Just test whether $$langle y,vrangle = lim_{ntoinfty} langle x_n,vrangle$$ holds for every $vin H$. This can be done by 2 steps. You test it for $v in text{span}{x_j;|;jgeq 1}$ first, and then for $v in text{span}^perp{x_j;|;jgeq 1}$. Since $$H = overline{text{span}}{x_j;|;jgeq 1} oplus text{span}^perp{x_j;|;jgeq 1},$$ this will prove the result.
answered Dec 7 '18 at 3:39
SongSong
10.8k628
$endgroup$
$begingroup$
I see that. Thanks a lot!
$endgroup$
– whereamI
Dec 7 '18 at 3:57
add a comment |
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1 Answer
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1 Answer
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$begingroup$
Just test whether $$langle y,vrangle = lim_{ntoinfty} langle x_n,vrangle$$ holds for every $vin H$. This can be done by 2 steps. You test it for $v in text{span}{x_j;|;jgeq 1}$ first, and then for $v in text{span}^perp{x_j;|;jgeq 1}$. Since $$H = overline{text{span}}{x_j;|;jgeq 1} oplus text{span}^perp{x_j;|;jgeq 1},$$ this will prove the result.
answered Dec 7 '18 at 3:39
SongSong
10.8k628
$endgroup$
$begingroup$
I see that. Thanks a lot!
$endgroup$
– whereamI
Dec 7 '18 at 3:57
add a comment |
$begingroup$
Just test whether $$langle y,vrangle = lim_{ntoinfty} langle x_n,vrangle$$ holds for every $vin H$. This can be done by 2 steps. You test it for $v in text{span}{x_j;|;jgeq 1}$ first, and then for $v in text{span}^perp{x_j;|;jgeq 1}$. Since $$H = overline{text{span}}{x_j;|;jgeq 1} oplus text{span}^perp{x_j;|;jgeq 1},$$ this will prove the result.
answered Dec 7 '18 at 3:39
SongSong
10.8k628
$endgroup$
$begingroup$
I see that. Thanks a lot!
$endgroup$
– whereamI
Dec 7 '18 at 3:57
add a comment |
$begingroup$
Just test whether $$langle y,vrangle = lim_{ntoinfty} langle x_n,vrangle$$ holds for every $vin H$. This can be done by 2 steps. You test it for $v in text{span}{x_j;|;jgeq 1}$ first, and then for $v in text{span}^perp{x_j;|;jgeq 1}$. Since $$H = overline{text{span}}{x_j;|;jgeq 1} oplus text{span}^perp{x_j;|;jgeq 1},$$ this will prove the result.
answered Dec 7 '18 at 3:39
SongSong
10.8k628
$endgroup$
Just test whether $$langle y,vrangle = lim_{ntoinfty} langle x_n,vrangle$$ holds for every $vin H$. This can be done by 2 steps. You test it for $v in text{span}{x_j;|;jgeq 1}$ first, and then for $v in text{span}^perp{x_j;|;jgeq 1}$. Since $$H = overline{text{span}}{x_j;|;jgeq 1} oplus text{span}^perp{x_j;|;jgeq 1},$$ this will prove the result.
answered Dec 7 '18 at 3:39
SongSong
10.8k628
edited Dec 7 '18 at 3:49
answered Dec 7 '18 at 3:39
SongSong
10.8k628
answered Dec 7 '18 at 3:39
SongSong
10.8k628
answered Dec 7 '18 at 3:39
SongSong
10.8k628
10.8k628
$begingroup$
I see that. Thanks a lot!
$endgroup$
– whereamI
Dec 7 '18 at 3:57
add a comment |
$begingroup$
I see that. Thanks a lot!
$endgroup$
– whereamI
Dec 7 '18 at 3:57
$begingroup$
I see that. Thanks a lot!
$endgroup$
– whereamI
Dec 7 '18 at 3:57
$begingroup$
I see that. Thanks a lot!
$endgroup$
– whereamI
Dec 7 '18 at 3:57
add a comment |
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