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MAST30026 Metric and Hilbert Spaces

创建日期：2024-06-04 17:31:56

所属分类：数学mathematics

标签： MAST30026

Metric and Hilbert Spaces

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MAST30026 Metric and Hilbert Spaces

Assignment 2
Note: Due Friday 13 October at 20:00 on Canvas & Gradescope. Please read the instructions
given on Canvas. The questions have varying lengths and do not all count for the same
number of marks; you may assume that longer questions are worth more.
1. Let U,V,W be normed spaces over F.
Suppose β ∶ U × V Ð→W is a continuous bilinear map.
Consider the linear function βU ∶ U Ð→ Hom(V,W ) given by βU(u) = fu, where
fu ∶ V Ð→W is defined by fu(v) = β(u, v).
(a) Prove that for any u ∈ U , fu ∈ B(V,W ), in other words fu is continuous.
(b) By part (a) we can think of βU as a function U Ð→ B(V,W ).
Prove that βU ∶ U Ð→ B(V,W ) is continuous.
2. In Proposition 3.23 we saw that the function
ℓ1 × ℓ∞ Ð→ F defined by (u, v)z→ ∞∑
n=1unvn
is a continuous bilinear map.
(a) Show that there is a continuous linear function ℓ1 Ð→ (c0)∨ that is an isometry.
(Recall that c0 ⊆ ℓ∞ consists of all convergent sequences with limit 0.)
[Hint : It may be useful to prove surjectivity first, and then the distance-preserving
property.]
(b) Conclude that ℓ1 is a Banach space.
(c) Where in your proof for (a) did you make use of the fact that you are working with c0
rather than ℓ∞?
3. Consider the maps Heven,Hodd ∶ FN Ð→ FN defined by
Heven((an)) = (a2n), Hodd((an)) = (a2n−1)
and construct f ∶ FN Ð→ FN × FN as
f(a) = (Heven(a),Hodd(a)).
(a) Prove that the restriction of Heven and Hodd to ℓp gives bounded linear functions
Heven,Hodd ∶ ℓp Ð→ ℓp for all p ∈ R⩾1 and for p =∞.
(b) Prove that f is an invertible linear map.
(c) Given p ∈ R⩾1, show that the restriction f ∶ ℓp Ð→ ℓp × ℓp is a linear isometry.
(Recall that we are working with the norm on ℓp × ℓp given by
∥(x, y)∥ ∶= ∥x∥ℓp + ∥y∥ℓp
as described in Example 3.4.)
(d) Show that the statement from part (b) does not hold for the space ℓ∞; prove the
strongest statement that you can for ℓ∞.
(Same comment as in part (c) applies for the norm we consider on ℓ∞ × ℓ∞.)
1
MAST30026 Metric and Hilbert Spaces 2023
4. Consider the map f ∶ ℓ1 Ð→ FN given by
f((an)) = (an
n
) .
(a) Prove that f maps to ℓ1 and f ∶ ℓ1 Ð→ ℓ1 is linear, continuous, and injective.
(b) Prove that the image W of f is not closed in ℓ1.
5. Let V = R2 viewed as a normed space with the Euclidean norm. Compute the norm of each
of the following elements M ∈ B(V,V ) directly from the description of the operator norm:
∥M∥ = sup∥v∥=1 ∥M(v)∥.
(a) A = (0 1
0 0
);
(b) B = ( 0 1−1 0);
(c) C = (a 0
0 b
) for a, b ∈ R.
6. We explore the Hilbert Projection Theorem when V is a Banach space but not a Hilbert
space.
(a) Let V = R2 with the ℓ1-norm, that is
∥(x1, x2)∥ = ∣x1∣ + ∣x2∣.
Let Y = B1(0), the closed unit ball around 0. Find two distinct closest points in Y to
x = (−1,1) ∈ V .
(b) Can you find a similar example for V = R2 with the ℓ∞-norm:
∥(x1, x2)∥ =max{∣x1∣, ∣x2∣}?
(c) Let V be a normed space and Y a convex subset of V . Fix x ∈ V . Let Z ⊆ Y be the
set of all closest points in Y to x. Prove that Z is convex.
7. Let H = ℓ2 over R and consider the subset
W = {y = (yn) ∈ ℓ2 ∶ yn ⩾ 0 for all n ∈ N}.
(a) Prove that W is a closed, convex subset of H. Is it a vector subspace?
(b) Find the closest point ymin ∈W to
x = (xn) = ((−1)n
n
) = (−1, 1
2
,−1
3
, . . .)
and compute dW (x).
[Hint : You may use without proof the identity
∞∑
n=1
1
n2
= π2
6
.]
2

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