
Operator in R^n
Let $\displaystyle A\in{B(R^n)}$ be linear invertible operator. Prove that $\displaystyle \forall\ x\in {R^n},\ \x\=1[$,
$\displaystyle
\exists i\in\{\pm1,\ldots,\pm n\}$, such that $\displaystyle \A^i(x)\\geq 1/n$.
Prove that constant $\displaystyle 1/n$ can't be increased. Prove also that if $\displaystyle S\subset\{\pm1,\ldots,\pm n\}$ is proper subset, then for
arbitrary $\displaystyle \varepsilon>0$ exists $\displaystyle A$ and $\displaystyle x$ so that $\displaystyle \A^i(x)\<\varepsilon,\ \forall\ i\in S$.