Homework 1 AMATH 563 solution

Description

Problems

Hint: read chapters 2 and 3 of Kreyszig.
1. Prove that 𝐶([𝑎, 𝑏]) equipped with the 𝐿
2
([𝑎, 𝑏]) norm is not a Banach space.

2. If (𝑋1, ‖ · ‖1) and (𝑋2, ‖ · ‖2) are normed spaces, show that the (Cartesian) product space 𝑋 = 𝑋1 × 𝑋2
becomes a normed space with the norm ‖𝑥‖ = max(‖𝑥1‖1, ‖𝑥2‖2) where 𝑥 ∈ 𝑋 is defined as the tuple
𝑥 = (𝑥1, 𝑥2) with addition and scalar multiplication operations: (𝑥1, 𝑥2) + (𝑦1, 𝑦2) = (𝑥1 + 𝑥2, 𝑦1 + 𝑦2)
and 𝛼(𝑥1, 𝑥2) = (𝛼𝑥1, 𝛼𝑥2).

3. Show that the product (composition) of two linear operators, if it exists, is a linear operator.

4. Let 𝑇 : 𝑋 → 𝑌 be a linear operator and dim𝑋 = dim𝑌 = 𝑛 < +∞. Show that Range(𝑇) = 𝑌 if and
only if 𝑇
−1
exists.

5. Let 𝑇 be a bounded linear operator from a normed space 𝑋 onto a normed space 𝑌 . Show that if there
is a positive constant 𝑏 such that ‖𝑇 𝑥‖ ≥ 𝑏‖𝑥‖ for all 𝑥 ∈ 𝑋 then 𝑇
−1
exists and is bounded.

6. Consider the functional 𝑓(𝑥) = max𝑡∈[𝑎,𝑏] 𝑥(𝑡) on 𝐶([𝑎, 𝑏]) equipped with the sup norm. Is this functional
linear? is it bounded?

7. Let 𝑋 be a Banach space and denote its dual as 𝑋*
. Show that ‖𝜙‖ : 𝜙 ↦→ sup‖𝑥‖=1 |𝜙(𝑥)| is a norm
on 𝑋*
.

8. Prove the Schwartz inequality on inner product spaces: |⟨𝑥, 𝑦⟩| ≤ ‖𝑥‖ · ‖𝑦‖ for all 𝑥, 𝑦 ∈ 𝑋, where
equality holds if and only if 𝑥, 𝑦 are linearly dependent.