Title page for etd-0621106-113021


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URN etd-0621106-113021
Author Ching-Jou Liao
Author's Email Address No Public.
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Department Applied Mathematics
Year 2005
Semester 2
Degree Master
Type of Document
Language English
Title Uniqueness of the norm preserving extension of a linear functional and the differentiability of the norm
Date of Defense 2006-06-16
Page Count 23
Keyword
  • norm preserving extension
  • property U
  • Abstract Let X be a Banach space and Y be a closed subspace of X. Given a
    bounded linear functional f on Y , the Hahn-Banach theorem guarantees
    that there exists a linear extension ˜ f 2 X of f which preserves the norm
    of f. But it does not state that such ˜ f is unique or not. If every f in Y 
    does have a unique norm preserving extension ˜ f in X , we say that Y has
    the unique extension property, or, following P. R. Phelps, the property U in
    X.
    A. E. Taylor [17] and S. R. Foguel [7] had shown that every subspace Y
    of X has the unique norm-preserving extension property in X if and only if
    the dual space X is strictly convex. As known in [11], X is smooth if X is
    strictly convex. The converse does not hold in general unless X is reflexive.
    In this thesis, we show that if a subspace Y of a Banach space X has
    the unique extension property then the norm of Y is outward smooth in X.
    The converse holds when Y is reflexive. Note that our conditions are local,
    i.e., they depend on Y only, but not on X. Several related results are also
    derived. Our work extends and unifies recent results in literature.
    Advisory Committee
  • Jen-Chih Yao - chair
  • Mark C. Ho - co-chair
  • Ying-Hsiung Lin - co-chair
  • none - co-chair
  • Ngai-Ching Wong - advisor
  • Files
  • etd-0621106-113021.pdf
  • indicate access worldwide
    Date of Submission 2006-06-21

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