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Unformatted text preview: Analysis 1
Miscellaneous Problems
1. Deﬁne a norm • on c0 by
∞ z=
n=0 zn
.
2n Is this norm complete? Is it equivalent to the sup norm • ∞? 2. State the HahnBanach Theorem. Prove that if X ( 0) and Y are normed spaces,
then completeness of L(X, Y ) forces completeness of Y .
3. Let Z be the space of all continuouslydiﬀerentiable realvalued functions on
[0, 1]. Decide whether Z is complete under the norm deﬁned by
f=f ∞ +f ∞. Is Z complete under the sup norm itself?
4. State the Closed Graph Theorem and deduce from it the Banach Isomophism
Theorem.
5. Let T : X → Y be a bounded linear map between Banach spaces. Prove that
if RanT is dense in Y and inf { T x : x = 1} > 0 then T is an isomorphism of
normed spaces.
6. Let X be a Banach space; let P : X → X be a linear map that is idempotent
(P ◦ P = P). Prove that P is bounded if and only if KerP and RanP are closed.
7. Deﬁne the quotient of a normed space X by its closed subspace Z . Prove that if
X/Z and Z are separable then so is X itself.
8. Let T ∈ L(X, Y ). Prove that if T is an open mapping, then completeness of X
implies that of Y .
9. Let (an )n∈N be a sequence of strictlypositive reals. Prove that there exists a
strictlypositive sequence (bn )n∈N with the properties
a n b 2 < ∞,
n
n∈N if and only if 1
n∈N an bn = ∞
n∈N = ∞.
1 10. Let B be a subset of the normed space Y . Show that B is bounded if and only
if ψ( B) is a bounded set of scalars for each ψ ∈ Y ∗ .
11. Let T ∈ L(X, Y ) be a bounded linear map between Banach spaces. Prove that
if it has closed range then
X/KerT RanT
(isomorphic as normed spaces).
12. Let T ∈ L(X, Y ) be a bounded linear map between normed spaces. Prove that
T ( Bo ) = Bo if and only if
X
Y
X/KerT ≡ Y
(isometrically isomorphic). 2 ...
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 Spring '09
 LARSON
 Linear map, Compact space, Banach space, Topological vector space, Banach Isomophism

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