Convergence in $ell^p$












1












$begingroup$



Let $0neq x=(x_1,x_2,cdots,x_n,cdots) inell^4$. For which of the following values of $p$, the series $$sum_{i=1}^infty x_iy_i$$ converges for every $y=(y_1,y_2,cdots,y_n,cdots) inell^p $ ?



(a) $p=1$ $hspace{2cm}$ (b) $p=2$$hspace{2cm}$ (c) $p=3$ $hspace{2cm}$ (d) $p=4$




I know this result: For any $a=(a_i) in ell^q$, the series $sum_{i=1}^infty a_iy_i$ converges absolutely, for all $y=(y_i) in ell^p$ where $q in (1,infty]$ and $p in [1,infty)$ with $frac{1}{p}+frac{1}{q}=1$



So by above result, $sum_{i=1}^infty x_iy_i$ converges absolutely, for all $y=(y_i) in largeell^frac{4}{3}$. But how about these four options ?



Here $$ sum_{i=1}^infty vert x_iy_i vert leq vertvert x vert vert_{largeell^4}vertvert y vert vert_{largeell^p} $$ How to choose suitable $p$ to make the RHS to be small? Any Guidance please?






convergence lp-spaces







share|cite|improve this question











$endgroup$












  • $begingroup$
    Recall that $ell^{p}subset ell^{4/3}$ for some values of $p$...which ones?
    $endgroup$
    – Giuseppe Negro
    Jan 21 at 10:25












  • $begingroup$
    You should try and refine your result by saying that the series converges for all $y in ell^p$ iff $p^{-1}+q^{-1} geq 1$.
    $endgroup$
    – Mindlack
    Jan 21 at 10:27










  • $begingroup$
    @GiuseppeNegro: $p leq 4/3$ . Thanks!
    $endgroup$
    – Chinnapparaj R
    Jan 21 at 10:32
















1












$begingroup$



Let $0neq x=(x_1,x_2,cdots,x_n,cdots) inell^4$. For which of the following values of $p$, the series $$sum_{i=1}^infty x_iy_i$$ converges for every $y=(y_1,y_2,cdots,y_n,cdots) inell^p $ ?



(a) $p=1$ $hspace{2cm}$ (b) $p=2$$hspace{2cm}$ (c) $p=3$ $hspace{2cm}$ (d) $p=4$




I know this result: For any $a=(a_i) in ell^q$, the series $sum_{i=1}^infty a_iy_i$ converges absolutely, for all $y=(y_i) in ell^p$ where $q in (1,infty]$ and $p in [1,infty)$ with $frac{1}{p}+frac{1}{q}=1$



So by above result, $sum_{i=1}^infty x_iy_i$ converges absolutely, for all $y=(y_i) in largeell^frac{4}{3}$. But how about these four options ?



Here $$ sum_{i=1}^infty vert x_iy_i vert leq vertvert x vert vert_{largeell^4}vertvert y vert vert_{largeell^p} $$ How to choose suitable $p$ to make the RHS to be small? Any Guidance please?






convergence lp-spaces







share|cite|improve this question











$endgroup$












  • $begingroup$
    Recall that $ell^{p}subset ell^{4/3}$ for some values of $p$...which ones?
    $endgroup$
    – Giuseppe Negro
    Jan 21 at 10:25












  • $begingroup$
    You should try and refine your result by saying that the series converges for all $y in ell^p$ iff $p^{-1}+q^{-1} geq 1$.
    $endgroup$
    – Mindlack
    Jan 21 at 10:27










  • $begingroup$
    @GiuseppeNegro: $p leq 4/3$ . Thanks!
    $endgroup$
    – Chinnapparaj R
    Jan 21 at 10:32














1












1








1





$begingroup$



Let $0neq x=(x_1,x_2,cdots,x_n,cdots) inell^4$. For which of the following values of $p$, the series $$sum_{i=1}^infty x_iy_i$$ converges for every $y=(y_1,y_2,cdots,y_n,cdots) inell^p $ ?



(a) $p=1$ $hspace{2cm}$ (b) $p=2$$hspace{2cm}$ (c) $p=3$ $hspace{2cm}$ (d) $p=4$




I know this result: For any $a=(a_i) in ell^q$, the series $sum_{i=1}^infty a_iy_i$ converges absolutely, for all $y=(y_i) in ell^p$ where $q in (1,infty]$ and $p in [1,infty)$ with $frac{1}{p}+frac{1}{q}=1$



So by above result, $sum_{i=1}^infty x_iy_i$ converges absolutely, for all $y=(y_i) in largeell^frac{4}{3}$. But how about these four options ?



Here $$ sum_{i=1}^infty vert x_iy_i vert leq vertvert x vert vert_{largeell^4}vertvert y vert vert_{largeell^p} $$ How to choose suitable $p$ to make the RHS to be small? Any Guidance please?






convergence lp-spaces







share|cite|improve this question











$endgroup$





Let $0neq x=(x_1,x_2,cdots,x_n,cdots) inell^4$. For which of the following values of $p$, the series $$sum_{i=1}^infty x_iy_i$$ converges for every $y=(y_1,y_2,cdots,y_n,cdots) inell^p $ ?



(a) $p=1$ $hspace{2cm}$ (b) $p=2$$hspace{2cm}$ (c) $p=3$ $hspace{2cm}$ (d) $p=4$




I know this result: For any $a=(a_i) in ell^q$, the series $sum_{i=1}^infty a_iy_i$ converges absolutely, for all $y=(y_i) in ell^p$ where $q in (1,infty]$ and $p in [1,infty)$ with $frac{1}{p}+frac{1}{q}=1$



So by above result, $sum_{i=1}^infty x_iy_i$ converges absolutely, for all $y=(y_i) in largeell^frac{4}{3}$. But how about these four options ?



Here $$ sum_{i=1}^infty vert x_iy_i vert leq vertvert x vert vert_{largeell^4}vertvert y vert vert_{largeell^p} $$ How to choose suitable $p$ to make the RHS to be small? Any Guidance please?






convergence lp-spaces




convergence lp-spaces






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 21 at 10:25







Chinnapparaj R

















asked Jan 21 at 10:22









Chinnapparaj RChinnapparaj R

5,6932928




5,6932928












  • $begingroup$
    Recall that $ell^{p}subset ell^{4/3}$ for some values of $p$...which ones?
    $endgroup$
    – Giuseppe Negro
    Jan 21 at 10:25












  • $begingroup$
    You should try and refine your result by saying that the series converges for all $y in ell^p$ iff $p^{-1}+q^{-1} geq 1$.
    $endgroup$
    – Mindlack
    Jan 21 at 10:27










  • $begingroup$
    @GiuseppeNegro: $p leq 4/3$ . Thanks!
    $endgroup$
    – Chinnapparaj R
    Jan 21 at 10:32


















  • $begingroup$
    Recall that $ell^{p}subset ell^{4/3}$ for some values of $p$...which ones?
    $endgroup$
    – Giuseppe Negro
    Jan 21 at 10:25












  • $begingroup$
    You should try and refine your result by saying that the series converges for all $y in ell^p$ iff $p^{-1}+q^{-1} geq 1$.
    $endgroup$
    – Mindlack
    Jan 21 at 10:27










  • $begingroup$
    @GiuseppeNegro: $p leq 4/3$ . Thanks!
    $endgroup$
    – Chinnapparaj R
    Jan 21 at 10:32
















$begingroup$
Recall that $ell^{p}subset ell^{4/3}$ for some values of $p$...which ones?
$endgroup$
– Giuseppe Negro
Jan 21 at 10:25






$begingroup$
Recall that $ell^{p}subset ell^{4/3}$ for some values of $p$...which ones?
$endgroup$
– Giuseppe Negro
Jan 21 at 10:25














$begingroup$
You should try and refine your result by saying that the series converges for all $y in ell^p$ iff $p^{-1}+q^{-1} geq 1$.
$endgroup$
– Mindlack
Jan 21 at 10:27




$begingroup$
You should try and refine your result by saying that the series converges for all $y in ell^p$ iff $p^{-1}+q^{-1} geq 1$.
$endgroup$
– Mindlack
Jan 21 at 10:27












$begingroup$
@GiuseppeNegro: $p leq 4/3$ . Thanks!
$endgroup$
– Chinnapparaj R
Jan 21 at 10:32




$begingroup$
@GiuseppeNegro: $p leq 4/3$ . Thanks!
$endgroup$
– Chinnapparaj R
Jan 21 at 10:32










1 Answer
1






active

oldest

votes


















2












$begingroup$

The correct answer is 1). Use the fact that $p leq frac 4 3 $ and $(y_n) in ell^{p}$ implies $(y_n) in ell^{4/3}$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thanks! So this question is a simple application of $ell^p subset ell^q$ whenever $p leq q$ and the mentioned result in OP. Am i right?
    $endgroup$
    – Chinnapparaj R
    Jan 21 at 10:32






  • 1




    $begingroup$
    @ChinnapparajR Yes, that is right.
    $endgroup$
    – Kavi Rama Murthy
    Jan 21 at 10:34











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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









2












$begingroup$

The correct answer is 1). Use the fact that $p leq frac 4 3 $ and $(y_n) in ell^{p}$ implies $(y_n) in ell^{4/3}$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thanks! So this question is a simple application of $ell^p subset ell^q$ whenever $p leq q$ and the mentioned result in OP. Am i right?
    $endgroup$
    – Chinnapparaj R
    Jan 21 at 10:32






  • 1




    $begingroup$
    @ChinnapparajR Yes, that is right.
    $endgroup$
    – Kavi Rama Murthy
    Jan 21 at 10:34
















2












$begingroup$

The correct answer is 1). Use the fact that $p leq frac 4 3 $ and $(y_n) in ell^{p}$ implies $(y_n) in ell^{4/3}$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thanks! So this question is a simple application of $ell^p subset ell^q$ whenever $p leq q$ and the mentioned result in OP. Am i right?
    $endgroup$
    – Chinnapparaj R
    Jan 21 at 10:32






  • 1




    $begingroup$
    @ChinnapparajR Yes, that is right.
    $endgroup$
    – Kavi Rama Murthy
    Jan 21 at 10:34














2












2








2





$begingroup$

The correct answer is 1). Use the fact that $p leq frac 4 3 $ and $(y_n) in ell^{p}$ implies $(y_n) in ell^{4/3}$






share|cite|improve this answer









$endgroup$



The correct answer is 1). Use the fact that $p leq frac 4 3 $ and $(y_n) in ell^{p}$ implies $(y_n) in ell^{4/3}$







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Jan 21 at 10:28









Kavi Rama MurthyKavi Rama Murthy

65.4k42766




65.4k42766












  • $begingroup$
    Thanks! So this question is a simple application of $ell^p subset ell^q$ whenever $p leq q$ and the mentioned result in OP. Am i right?
    $endgroup$
    – Chinnapparaj R
    Jan 21 at 10:32






  • 1




    $begingroup$
    @ChinnapparajR Yes, that is right.
    $endgroup$
    – Kavi Rama Murthy
    Jan 21 at 10:34


















  • $begingroup$
    Thanks! So this question is a simple application of $ell^p subset ell^q$ whenever $p leq q$ and the mentioned result in OP. Am i right?
    $endgroup$
    – Chinnapparaj R
    Jan 21 at 10:32






  • 1




    $begingroup$
    @ChinnapparajR Yes, that is right.
    $endgroup$
    – Kavi Rama Murthy
    Jan 21 at 10:34
















$begingroup$
Thanks! So this question is a simple application of $ell^p subset ell^q$ whenever $p leq q$ and the mentioned result in OP. Am i right?
$endgroup$
– Chinnapparaj R
Jan 21 at 10:32




$begingroup$
Thanks! So this question is a simple application of $ell^p subset ell^q$ whenever $p leq q$ and the mentioned result in OP. Am i right?
$endgroup$
– Chinnapparaj R
Jan 21 at 10:32




1




1




$begingroup$
@ChinnapparajR Yes, that is right.
$endgroup$
– Kavi Rama Murthy
Jan 21 at 10:34




$begingroup$
@ChinnapparajR Yes, that is right.
$endgroup$
– Kavi Rama Murthy
Jan 21 at 10:34


















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