Norms 1. Which of the following are norms? kxk = |x1 | + 7|x2 |, x R2 kxk = |x1 ||x2 |, Z 1 kf k = |f 0 (x)| dx x R2 f differentiable on (0, 1) 0 kf k = max |f (x)| f continuous on (0, 1) x[0,1] 2. Show that kxk kxk1 for all x Rn . 3. Calculate the condition number 2 (A) of the matrix \u0012 \u0013 1 1 A= . 1 1 4. Let kxk be a norm on Rn and kAk be the induced matrix norm for A Rnn . Show that (a) kABk kAkkBk for A, B Rnn , (b) kIk = 1 where I is the identity matrix, (c) (A) 1 Hint: Use the first two parts of this problem and I = AA1 . 1