Abstract
An (F)-spaceE is said to be locally midpoint constricted (in short, Imp-constricted) if there exists someδ>0 such thatD(A/2) <D(A) for every subsetA ofE with 0<D(A)≤δ, whereD(A) denotes the diameter ofA. Our main result goes as follow: LetE be an Imp-constricted (F)-space andU an open connected subset ofE. Assume thatT:U ⊸F is an isometry (i.e., a distance-preserving map) which mapsU onto an open subset of the (F)-spaceF. ThenT can be extended to an affine homeomorphism fromE toF. Also, some other results about the question whether each isometry between two (F)-spaces is affine are obtained.
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References
Charzyński, Z.,Sur les transformations isometrique des espaces du type (F), Studia Math.,13(1953), 94–121.
Day, M. M., Normed Linear Spaces, 3rd ed., Springer-Verlag, Berlin, Heidelberg, New York, 1973.
Dugunji, J., Topology, Allyn and Bacon, Inc., Boston, 1966.
Köthe, G., Topological Vector Spaces I, Springer-Verlag, Berlin, Heidelberg, New York, 1983.
Mankiewicz, P.,On extension of isometries in normed linear spaces, Bull. de I'Acad. Pol. des Sci., Ser. des Sci. Math., Astr. et Phys.,20(1972), 367–371.
Mazur, S. and Ulam, S.,Sur les transformations isométriques d'espaces vectoriels normés, Comp. Rend. Paris,194(1932), 946–948.
Rolewicz, S., Metric Linear Spaces, Polish Sci. Publ., Warsaw, 1972.
Vogt, A.,Maps which preserve equality of distance, Studia Math.,45(1973), 43–48.
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Guanggui, D., Senzhong, H. On extension of isometries in (F)-spaces. Acta Mathematica Sinica 12, 1–9 (1996). https://doi.org/10.1007/BF02109385
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DOI: https://doi.org/10.1007/BF02109385