Another approximation theoretic characterization of inner product spaces

Amir, Dan; Deutsch, Frank

doi:10.1090/S0002-9939-1978-0495846-6

Another approximation theoretic characterization of inner product spaces
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by Dan Amir and Frank Deutsch PDF

Proc. Amer. Math. Soc. 71 (1978), 99-102 Request permission

Abstract:

A normed space E is an inner product space if and only if for every 2-dimensional subspace V and every segment $I \subset V$, the corresponding metric projections satisfy the commutative property ${P_I}{P_V} = {P_V}{P_I}$.

References

Mahlon M. Day, Normed linear spaces, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21, Springer-Verlag, New York-Heidelberg, 1973. MR 0344849

Linear operators

R. A. Hirschfeld, On best approximations in normed vector spaces. II, Nieuw Arch. Wisk. (3) 6 (1958), 99–107. MR 106408
N. I. Gurariĭ and Ju. I. Sozonov, Normed spaces that do not have distortion of the unit sphere, Mat. Zametki 7 (1970), 307–310 (Russian). MR 264375
Robert C. James, Inner product in normed linear spaces, Bull. Amer. Math. Soc. 53 (1947), 559–566. MR 21242, DOI 10.1090/S0002-9904-1947-08831-5
J. T. Joichi, More characterizations of inner product spaces, Proc. Amer. Math. Soc. 19 (1968), 1185–1186. MR 236679, DOI 10.1090/S0002-9939-1968-0236679-9
E. R. Lorch, On certain implications which characterize Hilbert space, Ann. of Math. (2) 49 (1948), 523–532. MR 26249, DOI 10.2307/1969042
R. R. Phelps, Convex sets and nearest points, Proc. Amer. Math. Soc. 8 (1957), 790–797. MR 87897, DOI 10.1090/S0002-9939-1957-0087897-7
W. Rudin and K. T. Smith, Linearity of best approximation: a characterization of ellipsoids, Nederl. Akad. Wetensch. Proc. Ser. A 64 = Indag. Math. 23 (1961), 97–103. MR 0124717