Rotund renormings in spaces of Bochner integrable functions. (English) Zbl 1348.46010
The geometry of Köthe-Bochner spaces \(E\left( X\right) \) has been intensively developed during the last decades. The particular case of Lebesgue-Bochner spaces \(L^{p}\left( X\right) \) has often been considered first. Let \(L^{p}=L^{p}\left( T,\Sigma ,\mu \right)\). Recall that \( L^{p}\left( X\right) \) is the Banach space of all (equivalence classes of) Bochner integrable functions \(f:T\rightarrow X\) such that \(\left\| f\left( \cdot \right) \right\| _{X}\in L^{p}\) and it is endowed with the norm \(\left\| f\right\| _{L^{p}\left( X\right) }=\left\| \left\| f\left( \cdot \right) \right\| _{X}\right\| _{L^{p}}\).
The authors study basic geometric properties such as rotundity, uniform rotundity in every direction, local uniform rotundity and midpoint local uniform rotundity. It is known that these properties pass from a Banach space \(X\) to the space \(L^{p}\left( X\right) \) when \(1<p<\infty\). Clearly, this is not the case for \(L^{1}\left( X\right)\). The natural question is whether there is an equivalent norm \(\left| \left| \left| \cdot \right| \right| \right| \) on \(L^{1}\left( X\right) \) such that \(\left( L^{1}\left( X\right) ,\left| \left| \left| \cdot \right| \right| \right| \right) \) is rotund whenever \(X\) is. The authors answer this question affirmatively for rotundity and also for all above mentioned properties. The appropriate renorming is the Orlicz-Bochner norm associated to a suitable Orlicz function and the norm of \(X\).
The authors also prove that if \(X\) is uniformly rotund, then the norm \( \left| \left| \left| \cdot \right| \right| \right| \) on \(L^{1}\left( X\right) \) is such that its restriction to every reflexive subspace of \(L^{1}\left( X\right) \) is uniformly rotund.
The authors study basic geometric properties such as rotundity, uniform rotundity in every direction, local uniform rotundity and midpoint local uniform rotundity. It is known that these properties pass from a Banach space \(X\) to the space \(L^{p}\left( X\right) \) when \(1<p<\infty\). Clearly, this is not the case for \(L^{1}\left( X\right)\). The natural question is whether there is an equivalent norm \(\left| \left| \left| \cdot \right| \right| \right| \) on \(L^{1}\left( X\right) \) such that \(\left( L^{1}\left( X\right) ,\left| \left| \left| \cdot \right| \right| \right| \right) \) is rotund whenever \(X\) is. The authors answer this question affirmatively for rotundity and also for all above mentioned properties. The appropriate renorming is the Orlicz-Bochner norm associated to a suitable Orlicz function and the norm of \(X\).
The authors also prove that if \(X\) is uniformly rotund, then the norm \( \left| \left| \left| \cdot \right| \right| \right| \) on \(L^{1}\left( X\right) \) is such that its restriction to every reflexive subspace of \(L^{1}\left( X\right) \) is uniformly rotund.
Reviewer: Pawel Kolwicz (Poznań)
MSC:
46B03 | Isomorphic theory (including renorming) of Banach spaces |
46B20 | Geometry and structure of normed linear spaces |
46E40 | Spaces of vector- and operator-valued functions |