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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Integral equations
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Geometric Properties of Banach Spaces and Nonlinear Iterations (Paperback, 2009 ed.)
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Geometric Properties of Banach Spaces and Nonlinear Iterations (Paperback, 2009 ed.)
Series: Lecture Notes in Mathematics, 1965
Expected to ship within 10 - 15 working days
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The contents of this monograph fall within the general area of
nonlinear functional analysis and applications. We focus on an
important topic within this area: geometric properties of Banach
spaces and nonlinear iterations, a topic of intensive research
e?orts, especially within the past 30 years, or so. In this theory,
some geometric properties of Banach spaces play a crucial role. In
the ?rst part of the monograph, we expose these geometric
properties most of which are well known. As is well known, among
all in?nite dim- sional Banach spaces, Hilbert spaces have the
nicest geometric properties. The availability of the inner product,
the fact that the proximity map or nearest point map of a real
Hilbert space H onto a closed convex subset K of H is Lipschitzian
with constant 1, and the following two identities 2 2 2 ||x+y||
=||x|| +2 x,y +||y|| , (?) 2 2 2 2 ||?x+(1??)y|| = ?||x||
+(1??)||y|| ??(1??)||x?y|| , (??) which hold for all x,y? H, are
some of the geometric properties that char- terize inner product
spaces and also make certain problems posed in Hilbert spaces more
manageable than those in general Banach spaces. However, as has
been rightly observed by M. Hazewinkel, "... many, and probably
most, mathematical objects and models do not naturally live in
Hilbert spaces".
Consequently,toextendsomeoftheHilbertspacetechniquestomoregeneral
Banach spaces, analogues of the identities (?) and (??) have to be
developed.
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