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April 19th, 2017, 10:22 AM  #1 
Member Joined: Jun 2012 From: Uzbekistan Posts: 59 Thanks: 0  Inequality with max for independent vectors
Hello everybody. I have a problem which i can't prove or give a counterexample. Let $\displaystyle x=\{x_n\}_{n=1}^{\infty}$ and $\displaystyle y=\{x_n\}_{n=1}^{\infty}$ be two linearly independent vectors in infinite dimensional linear space i.e. $\displaystyle \{x_n\}_{n=1}^{\infty}\neq \{\lambda y_n\}_{n=1}^{\infty}$ with following properties: 1) $\displaystyle \max\limits_{1\leq n} \{x_n\} = \max\limits_{1\leq n} \{y_n\} = \max\limits_{1\leq n} \{a_{11}x_n+a_{12}y_n\} = \max\limits_{1\leq n} \{a_{21}x_n+a_{22}y_n\}=1$ 2) $\displaystyle \max\limits_{1\leq n} \{x_n+y_n\} =\max\limits_{1\leq n} \{(a_{11}+a_{21})x_n+(a_{12}+a_{22})y_n\}$ 3)$\displaystyle \max\limits_{1\leq n} \{x_ny_n\}=\max\limits_{1\leq n} \{(a_{11}a_{21})x_n+(a_{12}a_{22})y_n\}$. Where $\displaystyle a_{ij}$ real numbers such that $\displaystyle a_{11}a_{22}a_{12}a_{21}\neq 0$. Is it true that $\displaystyle a_{11}a_{22}a_{12}a_{21}\leq 1$? I could prove that the above is true in assumption $\displaystyle 1>a_{ij}>0$. For other cases i tried to find counterexample with help of Maple but since the conditions are many Maple couldn't solve a system of equations with max and absolute value. Can anybody help me with this problem? Thank's in advance. 

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independent, inequality, max, vectors 
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