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Miscellany

# Miscellany - Analysis 1 Miscellaneous Problems 1 Deﬁne a...

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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 Hahn-Banach 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 continuously-diﬀerentiable real-valued 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 strictly-positive reals. Prove that there exists a strictly-positive 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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Miscellany - Analysis 1 Miscellaneous Problems 1 Deﬁne a...

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