Math 2XX3
Practice Problems # 1
January 8, 2017, corrected Jan 12
Not to be submitted, but should be prepared for tutorial Jan 13-17.
1. Verify that the taxicab norm on R2 , k~xk = |x1 | + |x2 | satisfies the conditions which make
it a valid norm, that is:
(a) k~xk = 0 if and only if ~x = ~0;
(b) ka ~xk = |a| k~xk for all a ∈ R and ~x ∈ R2 ;
(c) k~x + ~y k ≤ k~xk + k~y k for all ~x, ~y ∈ R2 .
2. (a) Verify that the following identities hold for the Euclidean norm on Rn , defined by:
v
uX
√
u n 2
k~xk = ~x · ~x = t
xj
j=1
(i) [Paralellogram Law] k~x + ~y k2 + k~x − ~y k2 = 2k~xk2 + 2k~y k2 ;
1
k~x + ~y k2 − k~x − ~y k2 .
(ii) [Polarization Identity] ~x · ~y =
4
(b) Show that the Parallelogram Law becomes false if we replace the Euclidean norm on
R2 by the Taxicab norm (as defined in problem 1.)
3. Let U = {(x1 , x2 ) ∈ R2 : |x2 | ≤ x1 , and x1 > 0}.
Find all interior points of U and all boundary points of U . Is U an open set? Is U a closed
set?
4. Show each set is open, by showing every point ~a ∈ U is an interior point. [So you need
to explicitly find a radius r > 0 so that Dr (~a) ⊂ U .]
(a) U = {(x1 , x2 ) ∈ R2 : x21 + x22 > 0};
(b) U = {(x1 , x2 ) ∈ R2 : 1 < x21 + x22 < 4}.
5. Prove that:
x6
(a) lim 6 1 2 does not exist. (Note correction!)
~
x→~0 x1 + 3x2
x31 x22
= 0.
2
2 2
~
x→~0 [x1 + x2 ]
(b) lim
(c)
lim
~
x→(1,2)
(x1 − 1)2 (x2 − 2)2
3
[(x1 − 1)2 + (x2 − 2)2 ] 2
=0
1