Homework 1

Deadline: Oct. 7, 22:00:00

Norms, linear space and convergence

  1. We mentioned in the last lecture that in if converge to with one norm, then converge to with any norm.

    We now consider a special case, -norms.

    (1). Show that for any and any , .

    (2). Show that for any , if with norm then with norm .

Compact sets and extremum value

  1. Does have global maximum or minimum value on the set ?

Differential and gradients

  1. Compute the differentials of functions and .

    What is the relation between and , and what is the relation between and ?