Isomorophisms on $L^p$ and on $l^p$?












1












$begingroup$


I want to prove that $l^p$ is isomorphic to the infinite dierct sum of $l^p$, similarly for $L^p$. Every time I try to define an operator, I lose one of the properties that this operators must have like surjectivity or linearity !
Can you help me please !






functional-analysis







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$endgroup$












  • $begingroup$
    What topology do you want on the infinite direct sum? Or are you talking about isomorphism as vector spaces, without regard to topologies?
    $endgroup$
    – Mariano Suárez-Álvarez
    Mar 8 '13 at 2:24










  • $begingroup$
    The infinite direct product is not separable, @julien.
    $endgroup$
    – Mariano Suárez-Álvarez
    Mar 8 '13 at 2:28










  • $begingroup$
    @MarianoSuárez-Alvarez Sure, good point. What are talking about, here? I always find this terminology confusing. Is it the $C_0$ sum?
    $endgroup$
    – Julien
    Mar 8 '13 at 2:43










  • $begingroup$
    In our case we want the $ ||(x_n)||$ to be finite, were each coordinate of $(x_n)$ is an elements in $lp$, which means that the sequence of real numbers $(||x_n||)$ must belong to $lp$
    $endgroup$
    – user61965
    Mar 8 '13 at 3:17


















1












$begingroup$


I want to prove that $l^p$ is isomorphic to the infinite dierct sum of $l^p$, similarly for $L^p$. Every time I try to define an operator, I lose one of the properties that this operators must have like surjectivity or linearity !
Can you help me please !






functional-analysis







share|cite|improve this question











$endgroup$












  • $begingroup$
    What topology do you want on the infinite direct sum? Or are you talking about isomorphism as vector spaces, without regard to topologies?
    $endgroup$
    – Mariano Suárez-Álvarez
    Mar 8 '13 at 2:24










  • $begingroup$
    The infinite direct product is not separable, @julien.
    $endgroup$
    – Mariano Suárez-Álvarez
    Mar 8 '13 at 2:28










  • $begingroup$
    @MarianoSuárez-Alvarez Sure, good point. What are talking about, here? I always find this terminology confusing. Is it the $C_0$ sum?
    $endgroup$
    – Julien
    Mar 8 '13 at 2:43










  • $begingroup$
    In our case we want the $ ||(x_n)||$ to be finite, were each coordinate of $(x_n)$ is an elements in $lp$, which means that the sequence of real numbers $(||x_n||)$ must belong to $lp$
    $endgroup$
    – user61965
    Mar 8 '13 at 3:17
















1












1








1





$begingroup$


I want to prove that $l^p$ is isomorphic to the infinite dierct sum of $l^p$, similarly for $L^p$. Every time I try to define an operator, I lose one of the properties that this operators must have like surjectivity or linearity !
Can you help me please !






functional-analysis







share|cite|improve this question











$endgroup$




I want to prove that $l^p$ is isomorphic to the infinite dierct sum of $l^p$, similarly for $L^p$. Every time I try to define an operator, I lose one of the properties that this operators must have like surjectivity or linearity !
Can you help me please !






functional-analysis




functional-analysis






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 7 '18 at 3:49









Andrews

3671317




3671317










asked Mar 8 '13 at 2:10







user61965



















  • $begingroup$
    What topology do you want on the infinite direct sum? Or are you talking about isomorphism as vector spaces, without regard to topologies?
    $endgroup$
    – Mariano Suárez-Álvarez
    Mar 8 '13 at 2:24










  • $begingroup$
    The infinite direct product is not separable, @julien.
    $endgroup$
    – Mariano Suárez-Álvarez
    Mar 8 '13 at 2:28










  • $begingroup$
    @MarianoSuárez-Alvarez Sure, good point. What are talking about, here? I always find this terminology confusing. Is it the $C_0$ sum?
    $endgroup$
    – Julien
    Mar 8 '13 at 2:43










  • $begingroup$
    In our case we want the $ ||(x_n)||$ to be finite, were each coordinate of $(x_n)$ is an elements in $lp$, which means that the sequence of real numbers $(||x_n||)$ must belong to $lp$
    $endgroup$
    – user61965
    Mar 8 '13 at 3:17




















  • $begingroup$
    What topology do you want on the infinite direct sum? Or are you talking about isomorphism as vector spaces, without regard to topologies?
    $endgroup$
    – Mariano Suárez-Álvarez
    Mar 8 '13 at 2:24










  • $begingroup$
    The infinite direct product is not separable, @julien.
    $endgroup$
    – Mariano Suárez-Álvarez
    Mar 8 '13 at 2:28










  • $begingroup$
    @MarianoSuárez-Alvarez Sure, good point. What are talking about, here? I always find this terminology confusing. Is it the $C_0$ sum?
    $endgroup$
    – Julien
    Mar 8 '13 at 2:43










  • $begingroup$
    In our case we want the $ ||(x_n)||$ to be finite, were each coordinate of $(x_n)$ is an elements in $lp$, which means that the sequence of real numbers $(||x_n||)$ must belong to $lp$
    $endgroup$
    – user61965
    Mar 8 '13 at 3:17


















$begingroup$
What topology do you want on the infinite direct sum? Or are you talking about isomorphism as vector spaces, without regard to topologies?
$endgroup$
– Mariano Suárez-Álvarez
Mar 8 '13 at 2:24




$begingroup$
What topology do you want on the infinite direct sum? Or are you talking about isomorphism as vector spaces, without regard to topologies?
$endgroup$
– Mariano Suárez-Álvarez
Mar 8 '13 at 2:24












$begingroup$
The infinite direct product is not separable, @julien.
$endgroup$
– Mariano Suárez-Álvarez
Mar 8 '13 at 2:28




$begingroup$
The infinite direct product is not separable, @julien.
$endgroup$
– Mariano Suárez-Álvarez
Mar 8 '13 at 2:28












$begingroup$
@MarianoSuárez-Alvarez Sure, good point. What are talking about, here? I always find this terminology confusing. Is it the $C_0$ sum?
$endgroup$
– Julien
Mar 8 '13 at 2:43




$begingroup$
@MarianoSuárez-Alvarez Sure, good point. What are talking about, here? I always find this terminology confusing. Is it the $C_0$ sum?
$endgroup$
– Julien
Mar 8 '13 at 2:43












$begingroup$
In our case we want the $ ||(x_n)||$ to be finite, were each coordinate of $(x_n)$ is an elements in $lp$, which means that the sequence of real numbers $(||x_n||)$ must belong to $lp$
$endgroup$
– user61965
Mar 8 '13 at 3:17






$begingroup$
In our case we want the $ ||(x_n)||$ to be finite, were each coordinate of $(x_n)$ is an elements in $lp$, which means that the sequence of real numbers $(||x_n||)$ must belong to $lp$
$endgroup$
– user61965
Mar 8 '13 at 3:17












1 Answer
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$begingroup$

For $ell_p$, you want to consider breaking up the index set (presumably ${mathbb N} = {1,2,3,ldots}$ as the union of infinitely many infinite sets. For example, take a $2$-dimensional grid
$$ pmatrix{ 1 &2 &4 & 7 & ldots cr
3 &5 &8 & 12 & ldots cr
6 &9 &13 & 18 & ldots cr
10 & 14 & 19 & 25 & ldots cr
ldots &ldots & ldots & ldots & ldots cr}$$



and use the rows. Then $ell_p$ is the infinite direct sum of copies of $ell_p$ with the $p$-norm $|(X_1, X_2, ldots )|_p = left( sum_i |X_i|_p right)^p$, such that
$X = (x_1, x_2, x_3, ldots)$ corresponds to $(X_1, X_2, X_3, ldots)$ with
$X_1 = (x_1, x_2, x_4, x_7, ldots)$, $X_2 = (x_3, x_5, x_8, x_{12})$, ....



Similarly, for $L_p([0,1])$, break up the interval into countably many subintervals, say
$I_k = [a_k, a_{k+1}]$ where $a_n$ is an increasing sequence with $a_1 = 0$
and $lim_{n to infty} a_n = 1$.






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    $begingroup$

    For $ell_p$, you want to consider breaking up the index set (presumably ${mathbb N} = {1,2,3,ldots}$ as the union of infinitely many infinite sets. For example, take a $2$-dimensional grid
    $$ pmatrix{ 1 &2 &4 & 7 & ldots cr
    3 &5 &8 & 12 & ldots cr
    6 &9 &13 & 18 & ldots cr
    10 & 14 & 19 & 25 & ldots cr
    ldots &ldots & ldots & ldots & ldots cr}$$



    and use the rows. Then $ell_p$ is the infinite direct sum of copies of $ell_p$ with the $p$-norm $|(X_1, X_2, ldots )|_p = left( sum_i |X_i|_p right)^p$, such that
    $X = (x_1, x_2, x_3, ldots)$ corresponds to $(X_1, X_2, X_3, ldots)$ with
    $X_1 = (x_1, x_2, x_4, x_7, ldots)$, $X_2 = (x_3, x_5, x_8, x_{12})$, ....



    Similarly, for $L_p([0,1])$, break up the interval into countably many subintervals, say
    $I_k = [a_k, a_{k+1}]$ where $a_n$ is an increasing sequence with $a_1 = 0$
    and $lim_{n to infty} a_n = 1$.






    share|cite|improve this answer









    $endgroup$


















      3












      $begingroup$

      For $ell_p$, you want to consider breaking up the index set (presumably ${mathbb N} = {1,2,3,ldots}$ as the union of infinitely many infinite sets. For example, take a $2$-dimensional grid
      $$ pmatrix{ 1 &2 &4 & 7 & ldots cr
      3 &5 &8 & 12 & ldots cr
      6 &9 &13 & 18 & ldots cr
      10 & 14 & 19 & 25 & ldots cr
      ldots &ldots & ldots & ldots & ldots cr}$$



      and use the rows. Then $ell_p$ is the infinite direct sum of copies of $ell_p$ with the $p$-norm $|(X_1, X_2, ldots )|_p = left( sum_i |X_i|_p right)^p$, such that
      $X = (x_1, x_2, x_3, ldots)$ corresponds to $(X_1, X_2, X_3, ldots)$ with
      $X_1 = (x_1, x_2, x_4, x_7, ldots)$, $X_2 = (x_3, x_5, x_8, x_{12})$, ....



      Similarly, for $L_p([0,1])$, break up the interval into countably many subintervals, say
      $I_k = [a_k, a_{k+1}]$ where $a_n$ is an increasing sequence with $a_1 = 0$
      and $lim_{n to infty} a_n = 1$.






      share|cite|improve this answer









      $endgroup$
















        3












        3








        3





        $begingroup$

        For $ell_p$, you want to consider breaking up the index set (presumably ${mathbb N} = {1,2,3,ldots}$ as the union of infinitely many infinite sets. For example, take a $2$-dimensional grid
        $$ pmatrix{ 1 &2 &4 & 7 & ldots cr
        3 &5 &8 & 12 & ldots cr
        6 &9 &13 & 18 & ldots cr
        10 & 14 & 19 & 25 & ldots cr
        ldots &ldots & ldots & ldots & ldots cr}$$



        and use the rows. Then $ell_p$ is the infinite direct sum of copies of $ell_p$ with the $p$-norm $|(X_1, X_2, ldots )|_p = left( sum_i |X_i|_p right)^p$, such that
        $X = (x_1, x_2, x_3, ldots)$ corresponds to $(X_1, X_2, X_3, ldots)$ with
        $X_1 = (x_1, x_2, x_4, x_7, ldots)$, $X_2 = (x_3, x_5, x_8, x_{12})$, ....



        Similarly, for $L_p([0,1])$, break up the interval into countably many subintervals, say
        $I_k = [a_k, a_{k+1}]$ where $a_n$ is an increasing sequence with $a_1 = 0$
        and $lim_{n to infty} a_n = 1$.






        share|cite|improve this answer









        $endgroup$



        For $ell_p$, you want to consider breaking up the index set (presumably ${mathbb N} = {1,2,3,ldots}$ as the union of infinitely many infinite sets. For example, take a $2$-dimensional grid
        $$ pmatrix{ 1 &2 &4 & 7 & ldots cr
        3 &5 &8 & 12 & ldots cr
        6 &9 &13 & 18 & ldots cr
        10 & 14 & 19 & 25 & ldots cr
        ldots &ldots & ldots & ldots & ldots cr}$$



        and use the rows. Then $ell_p$ is the infinite direct sum of copies of $ell_p$ with the $p$-norm $|(X_1, X_2, ldots )|_p = left( sum_i |X_i|_p right)^p$, such that
        $X = (x_1, x_2, x_3, ldots)$ corresponds to $(X_1, X_2, X_3, ldots)$ with
        $X_1 = (x_1, x_2, x_4, x_7, ldots)$, $X_2 = (x_3, x_5, x_8, x_{12})$, ....



        Similarly, for $L_p([0,1])$, break up the interval into countably many subintervals, say
        $I_k = [a_k, a_{k+1}]$ where $a_n$ is an increasing sequence with $a_1 = 0$
        and $lim_{n to infty} a_n = 1$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Mar 8 '13 at 2:56









        Robert IsraelRobert Israel

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