Tensor product bases and tensor diagonals

Holub, J. R.

doi:10.1090/S0002-9947-1970-0279564-2

Tensor product bases and tensor diagonals
HTML articles powered by AMS MathViewer

by J. R. Holub PDF

Trans. Amer. Math. Soc. 151 (1970), 563-579 Request permission

Abstract:

Let X and Y denote Banach spaces with bases $({x_i})$ and $({y_i})$, respectively, and let $X{ \otimes _\varepsilon }Y$ and $X{ \otimes _\pi }Y$ denote the completion in the $\varepsilon$ and $\pi$ crossnorms of the algebraic tensor product $X \otimes Y$. The purpose of this paper is to study the structure of the tensor product spaces $X{ \otimes _\varepsilon }Y$ and $X{ \otimes _\pi }Y$ through a consideration of the properties of the tensor product basis $({x_i} \otimes {y_j})$ for these spaces and the tensor diagonal $({x_i} \otimes {y_i})$ of such bases.

References

C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151–164. MR 115069, DOI 10.4064/sm-17-2-151-164

Lectures on nuclear spaces and related topics

Paul Civin and Bertram Yood, Quasi-reflexive spaces, Proc. Amer. Math. Soc. 8 (1957), 906–911. MR 90020, DOI 10.1090/S0002-9939-1957-0090020-6

Normed linear spaces

26

Nelson Dunford and Robert Schatten, On the associate and conjugate space for the direct product of Banach spaces, Trans. Amer. Math. Soc. 59 (1946), 430–436. MR 15675, DOI 10.1090/S0002-9947-1946-0015675-0
Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
Ciprian Foiaş and Ivan Singer, On bases in $C([O,\,1])$ and $L^{1}\,([0,\,1])$, Rev. Roumaine Math. Pures Appl. 10 (1965), 931–960. MR 206691
B. R. Gelbaum and J. Gil de Lamadrid, Bases of tensor products of Banach spaces, Pacific J. Math. 11 (1961), 1281–1286. MR 147881
M. M. Grinblyum, On the representation of a space of type $B$ in the form of a direct sum of subspaces, Doklady Akad. Nauk SSSR (N.S.) 70 (1950), 749–752 (Russian). MR 0033977
Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
J. R. Holub, Tensor product bases and tensor diagonals, Trans. Amer. Math. Soc. 151 (1970), 563–579. MR 279564, DOI 10.1090/S0002-9947-1970-0279564-2
J. R. Holub and J. R. Retherford, Some curious bases for $c_{0}$ and $C[0,\,1]$, Studia Math. 34 (1970), 227–240. MR 271706, DOI 10.4064/sm-34-3-227-240

Schauder bases and topological tensor products

C. W. McArthur and J. R. Retherford, Some applications of an inequality in locally convex spaces, Trans. Amer. Math. Soc. 137 (1969), 115–123. MR 239391, DOI 10.1090/S0002-9947-1969-0239391-0
A. Pełczyński and W. Szlenk, An example of a non-shrinking basis, Rev. Roumaine Math. Pures Appl. 10 (1965), 961–966. MR 203432
J. R. Retherford, Shrinking bases in Banach spaces, Amer. Math. Monthly 73 (1966), 841–846. MR 201954, DOI 10.2307/2314177
J. R. Retherford, A semishrinking basis which is not shrinking, Proc. Amer. Math. Soc. 19 (1968), 766. MR 225144, DOI 10.1090/S0002-9939-1968-0225144-0
Helmut H. Schaefer, Topological vector spaces, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR 0193469
Robert Schatten, A Theory of Cross-Spaces, Annals of Mathematics Studies, No. 26, Princeton University Press, Princeton, N. J., 1950. MR 0036935
I. Singer, Basic sequences and reflexivity of Banach spaces, Studia Math. 21 (1961/62), 351–369. MR 146635, DOI 10.4064/sm-21-3-351-369
I. Singer, Weak$^{\ast }$ bases in conjugate Banach spaces. II, Rev. Math. Pures Appl. 8 (1963), 575–584. MR 178342
M. Zippin, A remark on bases and reflexivity in Banach spaces, Israel J. Math. 6 (1968), 74–79. MR 236677, DOI 10.1007/BF02771607