Prove that (bil(X × Y, Z), k · k) forms a normed vector space

1. Homework 7 Let X, Y, Z be normed vector spaces over R.

(1) Let bil(X × Y, Z) be the space of bounded bilinear maps from X × Y into Z. (For the definition of bounded bilinear maps, see class notes.) For each T ∈ bil(X × Y, Z), we define

kT k = sup

kxkX=kykY=1

kT (x, y)k_Z.

Prove that (bil(X × Y, Z), k · k) forms a normed vector space.

(2) A linear map φ : X → Y is called an isomorphism of normed vector spaces over R if φ is an isomorphism of vector spaces such that kφ(x)k_Y = kxk_X for any x ∈ X.

Prove that the map

ϕ : L^b(X, L^b(Y, Z)) → bil(X × Y, Z), T 7→ ϕ_T

defined by ϕ_T(x, y) = T (x)(y) is an isomorphism of normed vector spaces.

(3) Compute Df (0, 0), D²f (0, 0) and D³f (0, 0) for the following given functions f.

(a) f (x, y) = x⁴+ y⁴− x²− y²+ 1.

(b) f (x, y) = cos(x + 2y).

(c) f (x, y) = e^x+y.

(4) Let f : [a, b] → R be continuous. Suppose that Z b

a

f (x)h(x)dx = 0

for any continuous function h : [a, b] → R with h(a) = h(b) = 0. Show that f is the zero function on [a, b].

(5) Let C¹[a, b] be the space of all real valued continuously differentiable functions on [a, b]. On C¹[a, b], we define

kf k_C1 = kf k∞+ kf⁰k_∞ for any f ∈ C¹[a, b].

(a) Show that (C¹[a, b], k · k_C¹) is a Banach space over R.

(b) Let X be the subset of C¹[a, b] consisting of functions f : [a, b] → R so that f (a) = f (b) = 0. Prove that X forms a closed vector subspace of C¹[a, b]. Hence X is also a Banach space.

(6) Let L : U ⊆ R³→ R be a smooth function defined on an open subset of R³ and X_α,β = {f ∈ C¹[a, b] : f (a) = α, f (b) = β}

where α, β ∈ R. We consider the functional S : Xα,β → R defined by S(f ) =

Z b a

L(f (t), f⁰(t), t)dt.

Compute the Euler-Lagrange equation for S when L is given below.

(a) L(x, y, z) = sin y (b) L(x, y, z) = x²− y²

(c) L(x, y, z) = 2zy − y²+ 3x²y.

(7) Let f : [a, b] → R be a nonnegative continuously differentiable function on [a, b]. The area of the surface of revolution generated by rotating the curve y = f (x) about x-axis is

S(f ) = 2π Z b

a

f (x)p

1 + (f⁰(x))²dx.

1

2

Find the C¹-curve y = f (x) with f (a) = f (b) = 0 and f ≥ 0 on [a, b] minimizing S : X → R.

(8) A C¹-curve on R² is a map

γ : [0, 1] → R², t 7→ (x(t), y(t))

such that x(t), y(t) are C¹-functions on [0, 1]. If γ(0) = A and γ(1) = B, we say that γ is a curve from A to B. Here A, B are points of R². The arclength of γ is defined to be

L(γ) = Z 1

0

p(x⁰(t))²+ (y⁰(t))²dt.

Then L defines a real valued function on the space

P_A,B = {γ ∈ C¹([a, b], R²) : γ(0) = A, γ(1) = B}

of C¹-curve from A to B in R². Find the Euler Lagrange equation for L and prove that γ is the solution to the Euler-Lagrange equation for L if and only if γ the straight line connecting A and B.