Sum and product of all linear combinations












2














Let $K$ be a compact non-empty subset of $mathbb{R}^N$. If $x in mathbb{R}^N$ and if $(k_1, k_2, dots, k_N), k_i in mathbb{Z}_+$, consider the function
$$x mapsto x_1^{k_1}x_2^{k_2}...x_N^{k_N}.$$



I want to show that the set $mathcal{A}$ of all linear combinations of such functions is a $mathbb{R}$-subalgebra of $C(K)$, but I'm struggling specifically with proving that the sum and the product of elements of $mathcal{A}$ belongs to $mathcal{A}$ cause I've very much confused with manipulating the indexes.



$textbf{My attempt:}$
Consider the set
$$mathcal{A} = Big{ sum_{i=1}^{N}alpha_iprod_{i=1}^{N} x_i^{k_i}: x_i, alpha_i in mathbb{R}, k_i in mathbb{Z}_+ Big}$$
1. $p, q in mathcal{A} Rightarrow pq in mathcal{A}$



Take $p, q in mathcal{A}$ such that $p = sum_{i=1}^{N}alpha_iprod_{i=1}^{N} x_i^{k_i}$ and $q = sum_{j=1}^{N}tilde{alpha}_jprod_{j=1}^{N} x_j^{tilde{k}_j}$, then it follows:



begin{align*}
p(x)q(x) &= Big(sum_{i=1}^{N}alpha_iprod_{i=1}^{N} x_i^{k_i} Big) Big(sum_{j=1}^{N}tilde{alpha_j}prod_{j=1}^{N} x_j^{tilde{k}_j} Big) \
&= sum_{l=1}^{2N}c_l prod_{i=1}^{l}x_i^{k_i}, \
&= (pq)(x)
end{align*}

with $c_l = sum_{m=1}^{l}alpha_mtilde{alpha}_{l-m}$.




  1. $p, q in mathcal{A} Rightarrow p+q in mathcal{A}$


Take the same $p, q in mathcal{A}$ written above, then
begin{align*}
p(x) + q(x) &= Big(sum_{i=1}^{N}alpha_iprod_{i=1}^{N} x_i^{k_i} Big) + Big(sum_{j=1}^{N}tilde{alpha_j}prod_{j=1}^{N} x_j^{tilde{k}_j} Big) \
&= sum_{l=1}^{N} (alpha_l + tilde{alpha}_l)prod_{l=1}^{N}x_l^{k_l} \
&= (p + q)(x)
end{align*}










share|cite|improve this question





























    2














    Let $K$ be a compact non-empty subset of $mathbb{R}^N$. If $x in mathbb{R}^N$ and if $(k_1, k_2, dots, k_N), k_i in mathbb{Z}_+$, consider the function
    $$x mapsto x_1^{k_1}x_2^{k_2}...x_N^{k_N}.$$



    I want to show that the set $mathcal{A}$ of all linear combinations of such functions is a $mathbb{R}$-subalgebra of $C(K)$, but I'm struggling specifically with proving that the sum and the product of elements of $mathcal{A}$ belongs to $mathcal{A}$ cause I've very much confused with manipulating the indexes.



    $textbf{My attempt:}$
    Consider the set
    $$mathcal{A} = Big{ sum_{i=1}^{N}alpha_iprod_{i=1}^{N} x_i^{k_i}: x_i, alpha_i in mathbb{R}, k_i in mathbb{Z}_+ Big}$$
    1. $p, q in mathcal{A} Rightarrow pq in mathcal{A}$



    Take $p, q in mathcal{A}$ such that $p = sum_{i=1}^{N}alpha_iprod_{i=1}^{N} x_i^{k_i}$ and $q = sum_{j=1}^{N}tilde{alpha}_jprod_{j=1}^{N} x_j^{tilde{k}_j}$, then it follows:



    begin{align*}
    p(x)q(x) &= Big(sum_{i=1}^{N}alpha_iprod_{i=1}^{N} x_i^{k_i} Big) Big(sum_{j=1}^{N}tilde{alpha_j}prod_{j=1}^{N} x_j^{tilde{k}_j} Big) \
    &= sum_{l=1}^{2N}c_l prod_{i=1}^{l}x_i^{k_i}, \
    &= (pq)(x)
    end{align*}

    with $c_l = sum_{m=1}^{l}alpha_mtilde{alpha}_{l-m}$.




    1. $p, q in mathcal{A} Rightarrow p+q in mathcal{A}$


    Take the same $p, q in mathcal{A}$ written above, then
    begin{align*}
    p(x) + q(x) &= Big(sum_{i=1}^{N}alpha_iprod_{i=1}^{N} x_i^{k_i} Big) + Big(sum_{j=1}^{N}tilde{alpha_j}prod_{j=1}^{N} x_j^{tilde{k}_j} Big) \
    &= sum_{l=1}^{N} (alpha_l + tilde{alpha}_l)prod_{l=1}^{N}x_l^{k_l} \
    &= (p + q)(x)
    end{align*}










    share|cite|improve this question



























      2












      2








      2


      1





      Let $K$ be a compact non-empty subset of $mathbb{R}^N$. If $x in mathbb{R}^N$ and if $(k_1, k_2, dots, k_N), k_i in mathbb{Z}_+$, consider the function
      $$x mapsto x_1^{k_1}x_2^{k_2}...x_N^{k_N}.$$



      I want to show that the set $mathcal{A}$ of all linear combinations of such functions is a $mathbb{R}$-subalgebra of $C(K)$, but I'm struggling specifically with proving that the sum and the product of elements of $mathcal{A}$ belongs to $mathcal{A}$ cause I've very much confused with manipulating the indexes.



      $textbf{My attempt:}$
      Consider the set
      $$mathcal{A} = Big{ sum_{i=1}^{N}alpha_iprod_{i=1}^{N} x_i^{k_i}: x_i, alpha_i in mathbb{R}, k_i in mathbb{Z}_+ Big}$$
      1. $p, q in mathcal{A} Rightarrow pq in mathcal{A}$



      Take $p, q in mathcal{A}$ such that $p = sum_{i=1}^{N}alpha_iprod_{i=1}^{N} x_i^{k_i}$ and $q = sum_{j=1}^{N}tilde{alpha}_jprod_{j=1}^{N} x_j^{tilde{k}_j}$, then it follows:



      begin{align*}
      p(x)q(x) &= Big(sum_{i=1}^{N}alpha_iprod_{i=1}^{N} x_i^{k_i} Big) Big(sum_{j=1}^{N}tilde{alpha_j}prod_{j=1}^{N} x_j^{tilde{k}_j} Big) \
      &= sum_{l=1}^{2N}c_l prod_{i=1}^{l}x_i^{k_i}, \
      &= (pq)(x)
      end{align*}

      with $c_l = sum_{m=1}^{l}alpha_mtilde{alpha}_{l-m}$.




      1. $p, q in mathcal{A} Rightarrow p+q in mathcal{A}$


      Take the same $p, q in mathcal{A}$ written above, then
      begin{align*}
      p(x) + q(x) &= Big(sum_{i=1}^{N}alpha_iprod_{i=1}^{N} x_i^{k_i} Big) + Big(sum_{j=1}^{N}tilde{alpha_j}prod_{j=1}^{N} x_j^{tilde{k}_j} Big) \
      &= sum_{l=1}^{N} (alpha_l + tilde{alpha}_l)prod_{l=1}^{N}x_l^{k_l} \
      &= (p + q)(x)
      end{align*}










      share|cite|improve this question















      Let $K$ be a compact non-empty subset of $mathbb{R}^N$. If $x in mathbb{R}^N$ and if $(k_1, k_2, dots, k_N), k_i in mathbb{Z}_+$, consider the function
      $$x mapsto x_1^{k_1}x_2^{k_2}...x_N^{k_N}.$$



      I want to show that the set $mathcal{A}$ of all linear combinations of such functions is a $mathbb{R}$-subalgebra of $C(K)$, but I'm struggling specifically with proving that the sum and the product of elements of $mathcal{A}$ belongs to $mathcal{A}$ cause I've very much confused with manipulating the indexes.



      $textbf{My attempt:}$
      Consider the set
      $$mathcal{A} = Big{ sum_{i=1}^{N}alpha_iprod_{i=1}^{N} x_i^{k_i}: x_i, alpha_i in mathbb{R}, k_i in mathbb{Z}_+ Big}$$
      1. $p, q in mathcal{A} Rightarrow pq in mathcal{A}$



      Take $p, q in mathcal{A}$ such that $p = sum_{i=1}^{N}alpha_iprod_{i=1}^{N} x_i^{k_i}$ and $q = sum_{j=1}^{N}tilde{alpha}_jprod_{j=1}^{N} x_j^{tilde{k}_j}$, then it follows:



      begin{align*}
      p(x)q(x) &= Big(sum_{i=1}^{N}alpha_iprod_{i=1}^{N} x_i^{k_i} Big) Big(sum_{j=1}^{N}tilde{alpha_j}prod_{j=1}^{N} x_j^{tilde{k}_j} Big) \
      &= sum_{l=1}^{2N}c_l prod_{i=1}^{l}x_i^{k_i}, \
      &= (pq)(x)
      end{align*}

      with $c_l = sum_{m=1}^{l}alpha_mtilde{alpha}_{l-m}$.




      1. $p, q in mathcal{A} Rightarrow p+q in mathcal{A}$


      Take the same $p, q in mathcal{A}$ written above, then
      begin{align*}
      p(x) + q(x) &= Big(sum_{i=1}^{N}alpha_iprod_{i=1}^{N} x_i^{k_i} Big) + Big(sum_{j=1}^{N}tilde{alpha_j}prod_{j=1}^{N} x_j^{tilde{k}_j} Big) \
      &= sum_{l=1}^{N} (alpha_l + tilde{alpha}_l)prod_{l=1}^{N}x_l^{k_l} \
      &= (p + q)(x)
      end{align*}







      real-analysis linear-algebra polynomials






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      edited Dec 5 '18 at 19:03







      user71487

















      asked Nov 30 '18 at 19:47









      user71487user71487

      948




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