The isometric extension of the into mapping from the unit sphere S 1( E ) to S 1( l ∞(Γ)) |
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Authors: | Xiao Hong Fu |
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Institution: | (1) Department of Mathematics, Jiaying College, Meizhou, 514015, P. R. China |
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Abstract: | This paper considers the isometric extension problem concerning the mapping from the unit sphere S
1(E) of the normed space E into the unit sphere S
1(l
∞(Γ)). We find a condition under which an isometry from S
1(E) into S
1(l
∞(Γ)) can be linearly and isometrically extended to the whole space. Since l
∞(Γ) is universal with respect to isometry for normed spaces, isometric extension problems on a class of normed spaces are
solved. More precisely, if E and F are two normed spaces, and if V
0: S
1(E) → S
1(F) is a surjective isometry, where c
00(Γ) ⊆ F ⊆ l
∞(Γ), then V
0 can be extended to be an isometric operator defined on the whole space.
This work is supported by Natural Science Foundation of Guangdong Province, China (Grant No. 7300614) |
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Keywords: | l ∞ (Γ ) space isometric mapping isometric extension |
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