Homework 1 AMATH 563 solved

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Hint: read chapters 2 and 3 of Kreyszig.
1. Prove that 𝐢([π‘Ž, 𝑏]) equipped with the 𝐿
([π‘Ž, 𝑏]) norm is not a Banach space.
2. If (𝑋1, β€– Β· β€–1) and (𝑋2, β€– Β· β€–2) are normed spaces, show that the (Cartesian) product space 𝑋 = 𝑋1 Γ— 𝑋2
becomes a normed space with the norm β€–π‘₯β€– = max(β€–π‘₯1β€–1, β€–π‘₯2β€–2) where π‘₯ ∈ 𝑋 is defined as the tuple
π‘₯ = (π‘₯1, π‘₯2) with addition and scalar multiplication operations: (π‘₯1, π‘₯2) + (𝑦1, 𝑦2) = (π‘₯1 + π‘₯2, 𝑦1 + 𝑦2)
and 𝛼(π‘₯1, π‘₯2) = (𝛼π‘₯1, 𝛼π‘₯2).
3. Show that the product (composition) of two linear operators, if it exists, is a linear operator.
4. Let 𝑇 : 𝑋 β†’ π‘Œ be a linear operator and dim𝑋 = dimπ‘Œ = 𝑛 < +∞. Show that Range(𝑇) = π‘Œ if and
only if 𝑇
5. Let 𝑇 be a bounded linear operator from a normed space 𝑋 onto a normed space π‘Œ . Show that if there
is a positive constant 𝑏 such that ‖𝑇 π‘₯β€– β‰₯ 𝑏‖π‘₯β€– for all π‘₯ ∈ 𝑋 then 𝑇
exists and is bounded.
6. Consider the functional 𝑓(π‘₯) = maxπ‘‘βˆˆ[π‘Ž,𝑏] π‘₯(𝑑) on 𝐢([π‘Ž, 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 β€–πœ™β€– : πœ™ ↦→ supβ€–π‘₯β€–=1 |πœ™(π‘₯)| is a norm
on 𝑋*
8. Prove the Schwartz inequality on inner product spaces: |⟨π‘₯, π‘¦βŸ©| ≀ β€–π‘₯β€– Β· ‖𝑦‖ for all π‘₯, 𝑦 ∈ 𝑋, where
equality holds if and only if π‘₯, 𝑦 are linearly dependent.