Distinguished subspaces of $L_p$ of maximal dimension
Studia Mathematica, Tome 215 (2013) no. 3, pp. 261-280
Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences
Let $(\varOmega ,\varSigma ,\mu )$ be a measure space and $1 p \infty $. We show that, under quite general conditions, the set $L_{p}(\varOmega ) -
\bigcup _{1 \leq q p}L_{q}(\varOmega )$ is maximal spaceable, that is, it contains (except for the null vector) a closed subspace $F$ of $L_{p}(\varOmega )$ such that
$\operatorname{dim}(F) = \operatorname{dim}(L_{p}(\varOmega ))$. This result is so general that we had to develop a hybridization technique for measure spaces in order to construct a space such that the set $L_p(\varOmega ) - L_q(\varOmega )$, $1 \leq q p$, fails to be maximal spaceable. In proving these results we have computed the dimension of $L_p(\varOmega )$ for arbitrary measure spaces $(\varOmega ,\varSigma ,\mu )$. The aim of the results presented here is, among others, to generalize all the previous work (since the 1960's) related to the linear structure of the sets $L_{p}(\varOmega ) - L_{q}(\varOmega )$ with $q p$ and $L_{p}(\varOmega ) -
\bigcup_{1 \leq q p}L_{q}(\varOmega )$.
Keywords:
varomega varsigma measure space infty under quite general conditions set varomega bigcup leq varomega maximal spaceable contains except null vector closed subspace varomega operatorname dim operatorname dim varomega result general had develop hybridization technique measure spaces order construct space set varomega varomega leq fails maximal spaceable proving these results have computed dimension varomega arbitrary measure spaces varomega varsigma the results presented here among others generalize previous work since related linear structure sets varomega varomega varomega bigcup leq varomega
Affiliations des auteurs :
Geraldo Botelho 1 ; Daniel Cariello 1 ; Vinícius V. Fávaro 1 ; Daniel Pellegrino 2 ; Juan B. Seoane-Sepúlveda 3
@article{10_4064_sm215_3_4,
author = {Geraldo Botelho and Daniel Cariello and Vin{\'\i}cius V. F\'avaro and Daniel Pellegrino and Juan B. Seoane-Sep\'ulveda},
title = {Distinguished subspaces of $L_p$ of maximal dimension},
journal = {Studia Mathematica},
pages = {261--280},
year = {2013},
volume = {215},
number = {3},
doi = {10.4064/sm215-3-4},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/sm215-3-4/}
}
TY - JOUR AU - Geraldo Botelho AU - Daniel Cariello AU - Vinícius V. Fávaro AU - Daniel Pellegrino AU - Juan B. Seoane-Sepúlveda TI - Distinguished subspaces of $L_p$ of maximal dimension JO - Studia Mathematica PY - 2013 SP - 261 EP - 280 VL - 215 IS - 3 UR - http://geodesic.mathdoc.fr/articles/10.4064/sm215-3-4/ DO - 10.4064/sm215-3-4 LA - en ID - 10_4064_sm215_3_4 ER -
%0 Journal Article %A Geraldo Botelho %A Daniel Cariello %A Vinícius V. Fávaro %A Daniel Pellegrino %A Juan B. Seoane-Sepúlveda %T Distinguished subspaces of $L_p$ of maximal dimension %J Studia Mathematica %D 2013 %P 261-280 %V 215 %N 3 %U http://geodesic.mathdoc.fr/articles/10.4064/sm215-3-4/ %R 10.4064/sm215-3-4 %G en %F 10_4064_sm215_3_4
Geraldo Botelho; Daniel Cariello; Vinícius V. Fávaro; Daniel Pellegrino; Juan B. Seoane-Sepúlveda. Distinguished subspaces of $L_p$ of maximal dimension. Studia Mathematica, Tome 215 (2013) no. 3, pp. 261-280. doi: 10.4064/sm215-3-4
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