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Homework 1 AMATH 563 solved

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Problems

Hint: read chapters 2 and 3 of Kreyszig.
1. Prove that πΆ([π, π]) equipped with the πΏ
2
([π, π]) norm is not a Banach space.
2. If (π1, β Β· β1) and (π2, β Β· β2) are normed spaces, show that the (Cartesian) product space π = π1 Γ π2
becomes a normed space with the norm βπ₯β = max(βπ₯1β1, βπ₯2β2) where π₯ β π is defined as the tuple
π₯ = (π₯1, π₯2) with addition and scalar multiplication operations: (π₯1, π₯2) + (π¦1, π¦2) = (π₯1 + π₯2, π¦1 + π¦2)
and πΌ(π₯1, π₯2) = (πΌπ₯1, πΌπ₯2).
3. Show that the product (composition) of two linear operators, if it exists, is a linear operator.
4. Let π : π β π be a linear operator and dimπ = dimπ = π < +β. Show that Range(π) = π if and
only if π
β1
exists.
5. Let π be a bounded linear operator from a normed space π onto a normed space π . Show that if there
is a positive constant π such that βπ π₯β β₯ πβπ₯β for all π₯ β π then π
β1
exists and is bounded.
6. Consider the functional π(π₯) = maxπ‘β[π,π] π₯(π‘) on πΆ([π, π]) equipped with the sup norm. Is this functional
linear? is it bounded?
7. Let π be a Banach space and denote its dual as π*
. Show that βπβ : π β¦β supβπ₯β=1 |π(π₯)| is a norm
on π*
.
8. Prove the Schwartz inequality on inner product spaces: |β¨π₯, π¦β©| β€ βπ₯β Β· βπ¦β for all π₯, π¦ β π, where
equality holds if and only if π₯, π¦ are linearly dependent.