An n-Dimensional Hahn-Banach Extension Theorem and Minimal Projections |
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Authors: | BL Chalmers |
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Institution: | (1) Department of Mathematics, University of California, Riverside, California, 92521 |
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Abstract: | Let T~=i=1
n irvi:V V=v1,. . . .,vn] X, where i V* and X is a Banach space. Let T= i=1
nuivi: X V be an extension of T~ to all of X (i.e., ui X*) such that T has minimal (operator) norm. (E.g., if T~=I, T is a minimal projection from X onto V.) Then it is necessary and sufficient that u:=u_1,. . . ,un is given by (v:=v1,. . . ,vn)extv(u) Vn,where the notion of a v-extremal (extv) of u is properly defined.The condition above leads in many important cases to a simple geometric interpretation of minimal projections. Furthermore, by applying this formula to the case X=Lp, we obtain a linear n-dimensional analog of the Hölder equality condition (M is given by extv(u)=Mv)1/p u · Mv = 1/q u · Mv,wherever v is differentiable.We point out several applications, including the determination of the absolute projection constant of
n
p
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Keywords: | Hahn-Banach extensions minimal protections |
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