On moduli of convexity in Banach spaces
<p/> <p>Let <inline-formula><graphic file="1029-242X-2005-695306-i1.gif"/></inline-formula> be a normed linear space, <inline-formula><graphic file="1029-242X-2005-695306-i2.gif"/></inline-formula> an element of norm one, and <in...
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2005-01-01
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| Series: | Journal of Inequalities and Applications |
| Online Access: | http://www.journalofinequalitiesandapplications.com/content/2005/695306 |
| Summary: | <p/> <p>Let <inline-formula><graphic file="1029-242X-2005-695306-i1.gif"/></inline-formula> be a normed linear space, <inline-formula><graphic file="1029-242X-2005-695306-i2.gif"/></inline-formula> an element of norm one, and <inline-formula><graphic file="1029-242X-2005-695306-i3.gif"/></inline-formula> and <inline-formula><graphic file="1029-242X-2005-695306-i4.gif"/></inline-formula> the local modulus of convexity of <inline-formula><graphic file="1029-242X-2005-695306-i5.gif"/></inline-formula>. We denote by <inline-formula><graphic file="1029-242X-2005-695306-i6.gif"/></inline-formula> the greatest <inline-formula><graphic file="1029-242X-2005-695306-i7.gif"/></inline-formula> such that for each closed linear subspace <inline-formula><graphic file="1029-242X-2005-695306-i8.gif"/></inline-formula> of <inline-formula><graphic file="1029-242X-2005-695306-i9.gif"/></inline-formula> the quotient mapping <inline-formula><graphic file="1029-242X-2005-695306-i10.gif"/></inline-formula> maps the open <inline-formula><graphic file="1029-242X-2005-695306-i11.gif"/></inline-formula>-neighbourhood of <inline-formula><graphic file="1029-242X-2005-695306-i12.gif"/></inline-formula> in <inline-formula><graphic file="1029-242X-2005-695306-i13.gif"/></inline-formula> onto a set containing the open <inline-formula><graphic file="1029-242X-2005-695306-i14.gif"/></inline-formula>-neighbourhood of <inline-formula><graphic file="1029-242X-2005-695306-i15.gif"/></inline-formula> in <inline-formula><graphic file="1029-242X-2005-695306-i16.gif"/></inline-formula>. It is known that <inline-formula><graphic file="1029-242X-2005-695306-i17.gif"/></inline-formula>. We prove that there is no universal constant <inline-formula><graphic file="1029-242X-2005-695306-i18.gif"/></inline-formula> such that <inline-formula><graphic file="1029-242X-2005-695306-i19.gif"/></inline-formula>, however, such a constant <inline-formula><graphic file="1029-242X-2005-695306-i20.gif"/></inline-formula> exists within the class of Hilbert spaces <inline-formula><graphic file="1029-242X-2005-695306-i21.gif"/></inline-formula>. If <inline-formula><graphic file="1029-242X-2005-695306-i22.gif"/></inline-formula> is a Hilbert space with <inline-formula><graphic file="1029-242X-2005-695306-i23.gif"/></inline-formula>, then <inline-formula><graphic file="1029-242X-2005-695306-i24.gif"/></inline-formula>.</p> |
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| ISSN: | 1025-5834 1029-242X |