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Keywords:
reflexive Banach space; Schauder basis; quotient space; w$^*$-basic sequence; tensor product
reflexive Banach space; Schauder basis; quotient space; w$^*$-basic sequence; tensor product
Summary:
We observe that a separable Banach space $X$ is reflexive iff each of its quotients with Schauder basis is reflexive. Similarly if $\mathcal L(X,Y)$ is not reflexive for reflexive $X$ and $Y$ then $\mathcal L(X_1, Y)$ is is not reflexive for some $X_1\subset X$, $X_1$ having a basis.
References:
[1] W. J. Davis and J. Lindenstrauss: On total nonnorming subspaces. Proc. Amer. Math. Soc. 31 (1972), 109–111. DOI 10.1090/S0002-9939-1972-0288560-8 | MR 0288560
[2] J. Diestel: Sequences and Series in Banach Spaces. Springer-Verlag, Berlin-Heidelberg-New York, 1984. MR 0737004
[3] M. Fabian, P. Habala, P. Hájek, J. Pelant, V. Montesinos and V. Zizler: Functional Analysis and Infinite Dimensional Geometry. Canad. Math. Soc. Books in Mathematics Springer-Verlag, New York, 2001. MR 1831176
[4] S. Heinrich: On the reflexivity of the Banach space $L(X,Y)$. Funkts. Anal. Prilozh. 8 (1974), 97–98. (Russian) MR 0342991
[5] J. R. Holub: Reflexivity of $L(E,F)$. Proc. Amer. Math. Soc. 39 (1974), 175–177. MR 0315407
[6] H. Jarchow: Locally Convex Spaces. Teubner-Verlag, Stuttgart, 1981. MR 0632257 | Zbl 0466.46001
[7] W. B. Johnson and H. P. Rosenthal: On w$^*$ basic sequences and their applications to the study of Banach spaces. Studia Math. 43 (1972), 77–92. DOI 10.4064/sm-43-1-77-92 | MR 0310598
[8] J. Lindenstrauss and L. Tzafriri: Classical Banach Spaces I. Sequence Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete 92. Springer-Verlag, Berlin-Heidelberg-Berlin, 1977. MR 0500056
[9] J. Mujica: Reflexive spaces of homogeneous polynomials. Bull. Polish Acad. Sci. Math. 49 (2001), 211–222. MR 1863260 | Zbl 1068.46027
[10] A. Pełczyński: A note on the paper of I. Singer “Basic sequences and reflexivity of Banach spaces”. Studia Math. 21 (1962), 371–374. DOI 10.4064/sm-21-3-370-374 | MR 0146636
[11] V. Pták: Biorthogonal systems and reflexivity of Banach spaces. Czechoslovak Math. J. 9 (1959), 319–325. MR 0110008
[12] W. Ruckle: Reflexivity of $L(E,F)$. Proc. Am. Math. Soc. 34 (1972), 171–174. MR 0291777 | Zbl 0242.46018
[13] I. Singer: Basic sequences and reflexivity of Banach spaces. Studia Math. 21 (1962), 351–369. DOI 10.4064/sm-21-3-351-369 | MR 0146635 | Zbl 0114.30903
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