Rotundity in Lebesgue-Bochner function spaces

Smith, Mark A.; Turett, Barry

doi:10.1090/S0002-9947-1980-0549157-4

Rotundity in Lebesgue-Bochner function spaces
HTML articles powered by AMS MathViewer

by Mark A. Smith and Barry Turett PDF

Trans. Amer. Math. Soc. 257 (1980), 105-118 Request permission

Abstract:

This paper concerns the isometric theory of the Lebesgue-Bochner function space ${L^p}(\mu , X)$ where $1 < p < \infty$. Specifically, the question of whether a geometrical property lifts from X to ${L^p} (\mu , X)$ is examined. Positive results are obtained for the properties local uniform rotundity, weak uniform rotundity, uniform rotundity in each direction, midpoint local uniform rotundity, and B-convexity. However, it is shown that the Radon-Riesz property does not lift from X to ${L^p} (\mu , X)$. Consequently, Lebesgue-Bochner function spaces with the Radon-Riesz property are examined more closely.

References

Midpoint local uniform convexity, and other geometric properties of Banach spaces

Anatole Beck, A convexity condition in Banach spaces and the strong law of large numbers, Proc. Amer. Math. Soc. 13 (1962), 329–334. MR 133857, DOI 10.1090/S0002-9939-1962-0133857-9
S. Bochner and A. E. Taylor, Linear functionals on certain spaces of abstractly-valued functions, Ann. of Math. (2) 39 (1938), no. 4, 913–944. MR 1503445, DOI 10.2307/1968472
Mahlon M. Day, Some more uniformly convex spaces, Bull. Amer. Math. Soc. 47 (1941), 504–507. MR 4068, DOI 10.1090/S0002-9904-1941-07499-9
Mahlon M. Day, Strict convexity and smoothness of normed spaces, Trans. Amer. Math. Soc. 78 (1955), 516–528. MR 67351, DOI 10.1090/S0002-9947-1955-0067351-1
Mahlon M. Day, Normed linear spaces, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21, Springer-Verlag, New York-Heidelberg, 1973. MR 0344849
J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964
A. L. Garkavi, On the optimal net and best cross-section of a set in a normed space, Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962), 87–106 (Russian). MR 0136969
Daniel P. Giesy, On a convexity condition in normed linear spaces, Trans. Amer. Math. Soc. 125 (1966), 114–146. MR 205031, DOI 10.1090/S0002-9947-1966-0205031-7
Robert C. James, Uniformly non-square Banach spaces, Ann. of Math. (2) 80 (1964), 542–550. MR 173932, DOI 10.2307/1970663
I. E. Leonard, Banach sequence spaces, J. Math. Anal. Appl. 54 (1976), no. 1, 245–265. MR 420216, DOI 10.1016/0022-247X(76)90248-1
I. E. Leonard and K. Sundaresan, Geometry of Lebesgue-Bochner function spaces—smoothness, Bull. Amer. Math. Soc. 79 (1973), 546–549. MR 326376, DOI 10.1090/S0002-9904-1973-13195-7
A. R. Lovaglia, Locally uniformly convex Banach spaces, Trans. Amer. Math. Soc. 78 (1955), 225–238. MR 66558, DOI 10.1090/S0002-9947-1955-0066558-7
E. J. McShane, Linear functionals on certain Banach spaces, Proc. Amer. Math. Soc. 1 (1950), 402–408. MR 36448, DOI 10.1090/S0002-9939-1950-0036448-5

Theorie und Anwendungen der absolut additiven Mengen funktionen

122

Sur la convergence en moyenne

4

Haskell P. Rosenthal, Some applications of $p$-summing operators to Banach space theory, Studia Math. 58 (1976), no. 1, 21–43. MR 430749, DOI 10.4064/sm-58-1-21-43
Mark A. Smith, Products of Banach spaces that are uniformly rotund in every direction, Pacific J. Math. 70 (1977), no. 1, 215–219. MR 463892, DOI 10.2140/pjm.1977.73.215
Mark A. Smith, Some examples concerning rotundity in Banach spaces, Math. Ann. 233 (1978), no. 2, 155–161. MR 482080, DOI 10.1007/BF01421923
V. Šmulian, Sur la dérivabilité de la norme dans l’espace de Banach, C. R. (Doklady) Acad. Sci. URSS (N. S.) 27 (1940), 643–648 (French). MR 0002704
Kondagunta Sundaresan, Uniformly non-$l_{n}{}^{(1)}$ Orlicz spaces, Israel J. Math. 3 (1965), 139–146. MR 194876, DOI 10.1007/BF02759746
Kondagunta Sundaresan, Uniformly non-square Orlicz spaces, Nieuw Arch. Wisk. (3) 14 (1966), 31–39. MR 193497
Kondagunta Sundaresan, The Radon-Nikodým theorem for Lebesgue-Bochner function spaces, J. Functional Analysis 24 (1977), no. 3, 276–279. MR 0450956, DOI 10.1016/0022-1236(77)90058-1
Barry Turett and J. J. Uhl Jr., $L_{p}(\mu ,X)$ $(1<p<\infty )$ has the Radon-Nikodým property if $X$ does by martingales, Proc. Amer. Math. Soc. 61 (1976), no. 2, 347–350. MR 423069, DOI 10.1090/S0002-9939-1976-0423069-3
V. Zizler, On some rotundity and smoothness properties of Banach spaces, Dissertationes Math. (Rozprawy Mat.) 87 (1971), 33 pp. (errata insert). MR 300060