包含在$L_p(Ω) - \bigcup\limits_{q<p} L_q (Ω)$中的最大维数子空间
Subspaces of maximal dimension contained in $L_p(Ω) - \bigcup\limits_{q<p} L_q (Ω)$
摘要：
设$(Ω,Σ,μ)$是测度空间，$1< p < +\infty$ .在本文中，我们证明了，在相当一般的条件下，集合$L_{p}(Ω) - \bigcup\limits_{1 \leq q < p}L_{q}(Ω)$是极大可空的，也就是说，它包含（除了零向量）$L_{p}(Ω)$的一个闭子空间$F$，使得$\dim(F) = \dim(L_{p}(Ω))$。我们还表明，如果这些条件不满足，那么即使是更大的集合$L_p(Ω) - L_q(Ω)$，$1 \leq q < p$，可能无法最大空间。这里提出的结果的目的是，除其他外，推广所有以前的工作（自20世纪60年代）有关的线性结构的集合$L_{p}(Ω) - L_{q}(Ω)$与$q < p$和$L_{p}(Ω) - \bigcup\limits_{1 \leq q < p}L_{q}(Ω)$。
英文摘要：
Let $(Ω,Σ,μ)$ be a measure space and $1< p < +\infty$. In this paper we show that, under quite general conditions, the set $L_{p}(Ω) - \bigcup\limits_{1 \leq q < p}L_{q}(Ω)$ is maximal spaceable, that is, it contains (except for the null vector) a closed subspace $F$ of $L_{p}(Ω)$ such that $\dim(F) = \dim(L_{p}(Ω))$. We also show that if those conditions are not fulfilled, then even the larger set $L_p(Ω) - L_q(Ω)$, $1 \leq q < p$, may fail to be maximal spaceable. 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 $L_{p}(Ω) - L_{q}(Ω)$ with $q < p$ and $L_{p}(Ω) - \bigcup\limits_{1 \leq q < p}L_{q}(Ω)$.
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