(b) Prove that kvk∞≤ kvk2≤ kvk1

2. HW7 2.1. Part I.

(1) On R², we define functions

k · k1, k · k2, k · k_∞: R²→ [0, ∞).

as follows. For v = (a, b) ∈ R², we set kvk1= |a| + |b|, kvk2=p

a²+ b², kvk∞= max{|a|, |b|}.

(a) Prove that these functions define norms on R². (b) Prove that kvk∞≤ kvk2≤ kvk1.

(c) Sketch the unit circles in (R², k · k1) and in (R², k · k2) and in (R², k · k∞). In other words, graph

{v ∈ R²: kvk1= 1}, {v ∈ R²: kvk2= 1}, {v ∈ R²: kvk∞= 1}.

(2) An inner product on Rⁿ is a function

h·, ·i : Rⁿ× Rⁿ → [0, ∞) such that for u, v, w ∈ Rⁿ and a ∈ R,

(i) hv, wi = hw, vi,

(ii) hv, vi = 0 if and only if v = 0, (iii) hu + v, wi = hu, wi + hv, wi.

(iii) hau, vi = ahu, vi.

(a) Let a = (a₁, a₂) be a vector in R²such that a₁, a₂ are positive real numbers. Suppose that u = (u1, u2) and v = (v1, v2) are vectors in R². We define

hu, via= a₁u₁v₁+ a₂u₂v₂.

Show that h·, ·ia defines an inner product on R². (When a1 = a2 = 1, we obtain the Euclidean inner product on R².)

(b) Suppose h·, ·i is an inner product (not necessarily the inner product h·, ·i_a defined in (a)) on R². We define

kvk =phv, vi.

Show that k · k defines a norm on R². This norm is called the normed induced from the inner product h·, ·i.

(c) As in (b), if k · k is the norm induced from the inner product h·, ·i, prove that the norm satisfies the parallelogram law:

kv + wk²+ kv − wk²= 2kvk²+ 2kwk² and the polarization identity:

4hv, wi = kv + wk²− kv − wk².

(d) Let h·, ·i be an inner product on R². Prove that the Cauchy-Schwarz inequality holds:

|hv, wi| ≤ kvkkwk, v, w ∈ Rⁿ. Hint: Consider the function f (t) = ktv − wk² for t ∈ R.

(e) Prove that the parallelogram law does not hold for the norm k · k1 and for the norm k · k_∞ on R².

(3) Let {v1, · · · , vk} be a set of vectors in the n-dimensional Euclidean space Rⁿ. The Gram determinant of {v1, · · · , vk} is defined to be

Γ(v1, v2, · · · , vk) = det[hvi, vji]^k_i,j=1.

The volume of the k-dimensional parallelepiped P spanned by {v₁, · · · , v_k} equals vol (P ) =p

Γ(v1, · · · , vk).

1

2

(a) Let v = (1, 1, 1) and w = (1, 0, 1) be vectors in R³. (The set {v, w} is linearly indepen- dent over R.)

(i) Use the formulapkvk²kwk²− hv, wi² to calculate the area of the parallelogram spanned by {v, w}.

(ii) Let M =



 1 1 1 0 1 1



be a 3 × 2 matrix whose column vectors consisting of v, w.

Find M^TM where M^T is the transpose of M. Calculate Γ(v, w) using the fact that the determinant of M^TM is exactly the Gram determinant of {v, w}. Verify thatpΓ(v, w) equals the answer obtained in (i).

(b) Let u = (2, 0, 1, 1, 0) and v = (0, 0, 2, 1, 2) and w = (1, 1, 1, 1, 1) be vectors in the five dimensional Euclidean space R⁵. (They are linearly independent over R.) Calculate the volume of the parallelepiped:

P = {au + bv + cw : a, b, c ∈ [0, 1]}

spanned by {u, v, w} using the Gram determinant Γ(u, v, w). Hint: consider the 5 × 3

matrix M =













2 0 1 0 0 1 1 2 1 1 1 1 0 2 1













. Compute M^TM and its determinant.

2.2. Part II.

(1) Sketch the domain D of the function f. Here D is the maximum subset of R² consisting of points (x, y) so that f (x, y) is defined.

(a) f (x, y) =√ y +p

25 − x²− y². (b) f (x, y) = sin⁻¹(x²+ y²− 2).

(2) Sketch the graph of the functions. The domain of the function f is the maximum subset D consisting of (x, y) so that f (x, y) is defined.

(a) f (x, y) = 10 − 4x − 5y.

(b) f (x, y) =p

4 − 4x²− y².

(3) Prove that the following subsets of R² are open.

(a) R²\ {(0, 0)}.

(b) {(x, y) ∈ R²: xy > 1}.

(4) Is it true that int(A) ∩ int(B) = int(A ∩ B)? Here A, B are subsets of R². Prove or disprove.

(5) Show that any finite subset of R² is closed.

(6) Let S = {(x, y) ∈ R²: x²+ y²≤ 1}. Is S closed? Prove or disprove.

(7) Let S = {(x, y) ∈ R²: xy ≥ 1}. Find int(S). (You need to prove that your answer is correct.) (8) Find the accumulation points of A = {(x, y) ∈ R² : y = 0, 0 < x < 1}. (You need to prove

that your answer is correct.)

(9) Find the closure A of A = {(x, y) ∈ R² : x > y²}. (You need to prove that your answer is correct.)

(10) Find the ∂A boundary of A = {(x, y) ∈ R² : x²− y² > 1}. (You need to prove that your answer is correct.)

(11) Find the limits.

(a) lim

(x,y)→(1,0)

x sin y x²+ 1. (b) lim

(x,y)→(0,0)

cos x²+ y³ x + y + 1. (c) lim

(x,y)→(2,−4)

y + 4

x²y − xy + 4x²− 4x.

3

(d) lim

(x,y)→(4,3)

√x −√ y + 1 x − y − 1 .

(12) Show that the functions below has or has no limit as (x, y) → (0, 0).

(a) f (x, y) = xy

|xy|.

(b) f (x, y) = tan⁻¹ |x| + |y|

x²+ y²



. (use Polar coordinate)