Difference between revisions of "Tsirelson space"
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− | A specific example of a reflexive [[Banach space|Banach space]] (cf. [[Reflexive space|Reflexive space]]) which does not contain an imbedded | + | {{TEX|done}} |
+ | A specific example of a reflexive [[Banach space|Banach space]] (cf. [[Reflexive space|Reflexive space]]) which does not contain an imbedded $l_p$-space or an imbedded $c_0$-space. On the other hand, the classical Banach spaces, such as the spaces $L_p(\mu)=L_p(\Omega,\Sigma,\mu)$ of equivalence classes of measurable functions whose $p$-th powers are integrable and the spaces $C(K)$ of continuous scalar-valued functions on $K$ with the supremum norm, all do contain a copy of $c_0$ or $l_p$, and so do all Orlicz spaces (cf. [[Orlicz space|Orlicz space]]). | ||
− | For a selection of results concerning Banach spaces which do contain | + | For a selection of results concerning Banach spaces which do contain $l_p$ or $c_0$ see [[#References|[a3]]], Sect. 2e. |
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B.S. Tsirelson, "Not every Banach space contains an imbedding of | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B.S. Tsirelson, "Not every Banach space contains an imbedding of $l_p$ or $c_0$" ''Funct. Anal. Appl.'' , '''8''' : 2 (1974) pp. 138–141 ''Funkts. Anal. Prilozhen.'' , '''8''' : 2 (1974) pp. 57–60</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P.G. Casazza, Th.J. Shura, "Tsirelson's space" , ''Lect. notes in math.'' , '''1363''' , Springer (1989)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J. Lindenstrauss, L. Tzafriri, "Classical Banach spaces" , '''1. Sequence spaces''' , Springer (1977)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> D. van Dulst, "Characterizations of Banach spaces not containing $L^1$" , CWI (1989)</TD></TR></table> |
Latest revision as of 14:06, 11 August 2014
A specific example of a reflexive Banach space (cf. Reflexive space) which does not contain an imbedded $l_p$-space or an imbedded $c_0$-space. On the other hand, the classical Banach spaces, such as the spaces $L_p(\mu)=L_p(\Omega,\Sigma,\mu)$ of equivalence classes of measurable functions whose $p$-th powers are integrable and the spaces $C(K)$ of continuous scalar-valued functions on $K$ with the supremum norm, all do contain a copy of $c_0$ or $l_p$, and so do all Orlicz spaces (cf. Orlicz space).
For a selection of results concerning Banach spaces which do contain $l_p$ or $c_0$ see [a3], Sect. 2e.
References
[a1] | B.S. Tsirelson, "Not every Banach space contains an imbedding of $l_p$ or $c_0$" Funct. Anal. Appl. , 8 : 2 (1974) pp. 138–141 Funkts. Anal. Prilozhen. , 8 : 2 (1974) pp. 57–60 |
[a2] | P.G. Casazza, Th.J. Shura, "Tsirelson's space" , Lect. notes in math. , 1363 , Springer (1989) |
[a3] | J. Lindenstrauss, L. Tzafriri, "Classical Banach spaces" , 1. Sequence spaces , Springer (1977) |
[a4] | D. van Dulst, "Characterizations of Banach spaces not containing $L^1$" , CWI (1989) |
How to Cite This Entry:
Tsirelson space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tsirelson_space&oldid=14244
Tsirelson space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tsirelson_space&oldid=14244