Homework 1

Due: Sept. 30, 14:00:00

Norms, linear space and convergence

We would like to show that in a finite dimensional linear space, if {xn} converge to x with respect to one norm, then {xn} converge to x with respect to any norm.

Suppose that V is a finite dimensional linear space (you can assume V=Rn), a and b are two norms.

  1. Show that b:VR is a bounded function on the set {v:va=1}.
  2. Show that b:VR is a continuous function with respect to norm a. Namely, for any vV, and any ε>0, there exists δ>0 such that for all uV, if vua<δ then vub<ε.

Then it is a simple corollary that {xn}x in norm a implies {xn}x in norm b.

Compact sets and extremum value

  1. Does f(x1,x2,x3)=3x13+x1x2+ex3+ln|x1+x2x3| have extreme value on the set {(x1,x2,x3):x12+2x1x2+2x22x1x32x2x3+x3210} ?

Differential and gradients

  1. Compute the differentials of functions g(r,θ,φ)=(rsinθcosφ,rsinθsinφ,rcosθ)T and h(x,y,z)=(x2+y2+z2,arccoszx2+y2+z2,arctanyx)T.

    What is the relationship between Dh and Dg ?