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April 19th, 2017, 10:22 AM   #1
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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)$\displaystyle \max\limits_{1\leq n} \{|x_n-y_n|\}=\max\limits_{1\leq n} \{|(a_{11}-a_{21})x_n+(a_{12}-a_{22})y_n|\}$.

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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