Problem Set - Absolute Value

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 7 December 2009

Absolute value

	Let $x\in ℝ$. Define $\left|x\right|$.
	Let $x\in ℂ$. Define $\left|x\right|$.
	Let $x\in ℝ$. Show that $\left|x\right|=|x+0i|$.
	Let $x\in ℝ$. Show that $|-x|=\left|x\right|$.
	Let $x,y\in ℝ$. Show that $|x+y|\le \left|x\right|+\left|y\right|$.
	Let $x,y\in ℂ$. Show that $|x+y|\le \left|x\right|+\left|y\right|$.
	Let $x,y,z\in ℝ$. Show that $|x+y+z|\le \left|x\right|+\left|y\right|+\left|z\right|$.
	Let $x,y,z\in ℂ$. Show that $|x+y+z|\le \left|x\right|+\left|y\right|+\left|z\right|$.
	Let $x,y\in ℂ$. Show that ${|x+y|}^2+{|x-y|}^2=2({\left|x\right|}^2+{\left|y\right|}^2)$.
	Let $x,y\in ℂ$. Show that ${|x+y|}^2={\left|x\right|}^2+{\left|y\right|}^2+2\mathrm{Re}\left(a\overline{b}\right)$.
	Let $x,y\in ℝ$. Show that $|x+y|\ge \left|\text{<mi>\hspace{0.17em}</mi>}\right|x|-|y\left|\text{<mi>\hspace{0.17em}</mi>}\right|$.
	Let $x,y\in ℝ$. Show that $|x-y|\ge \left|\text{<mi>\hspace{0.17em}</mi>}\right|x|-|y\left|\text{<mi>\hspace{0.17em}</mi>}\right|$.
	Let $x,y,z\in ℝ$. Show that $|x+y+z|\ge \left|\text{<mi>\hspace{0.17em}</mi>}\right|x|-|y|-|z\left|\text{<mi>\hspace{0.17em}</mi>}\right|$.
	Give solutions to the following inequalities in terms of intervals: (a)   $\left|x\right|>3$. (b)   $|1+2x|\le 4$. (c)   $|x+2|\ge 5$. (d)   $|x-5|<|x+1|$. (e)   $|x-2|<3$ or $|x+1|<1$. (f)   $|x-2|<3$ and $|x+1|<1$.
	Let $a,b\in ℝ$ and let $0<\epsilon <\left|b\right|$. Show that ${\displaystyle \left|\frac{a+\epsilon }{b+\epsilon }\right|\le \frac{\left|a\right|+\epsilon }{\left|b\right|+\epsilon }}$.
	Prove that if $a_1,a_2,\dots ,a_n\in ℝ$ then ${\displaystyle \left|\sum _{k=1}^n{a}_k\right|}\le {\displaystyle \sum _{k=1}^n\left|a_k\right|}$.
	Prove that if $a_1,a_2,\dots ,a_n\in ℝ$ then ${\displaystyle \left|\sum _{k=1}^n{a}_k\right|}\ge \left|a_p\right|-{\displaystyle \sum _{k=1,k\ne p}^n\left|a_k\right|}$.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)