arXiv 2108.006282021-08math.FAmath.GN

关于巴拿赫空间中的性质- $(P_{1})$

On Property-$(P_{1})$ in Banach spaces

Teena Thomas

备注：: Minor modifications, results unchanged, to appear in Journal of Convex Analysis

摘要：

我们讨论了J. Mach [Chebyshev中心的连续性性质，J. Approx.理论，29（3）：223- 230，1980]作为性质- $(P_1)$。本文证明了：若Banach空间$X$的闭子空间的闭单位球对$X$的闭有界子集、紧子集和有限子集的每一类都具有性质- $(P_1)$，则该子空间也具有性质.证明了$L_{1}$ -预对偶空间中的$M$ -理想的闭单位球对于空间的紧子集满足性质- $(P_{1})$.对于Choquet单纯形$K$，给出了$A(K)$的有限余维闭子空间的闭单位球满足$A(K)$的紧子集的性质- $(P_{1})$的充分条件.这个条件也有助于建立$A(K)$的有限余维子空间的闭单位球的强逼近性与$A(K)$的紧子集的子空间的闭单位球的性质- $(P_{1})$的等价性.进一步，对于紧Hausdorff空间$S$，利用子空间的性质- $(P_{1})$及其闭单位球对$C(S)$的紧子集的性质- $(P_{1})$，给出了$C(S)$的强逼近有限余维闭子空间的一个特征.我们将这一刻画推广到$L_{1}$ -预对偶空间的强逼近有限余维闭子空间。因此，我们证明了这样的子空间是超平面的有限交，使得每个超平面的闭单位球对于$L_1$ -预对偶空间的紧子集满足性质- $(P_1)$，反之亦然。最后，给出了一个非自反Banach空间的闭子空间满足$1 \frac{1}{2}$ -球性质且不允许有界闭子集有限制Chebyshev中心的例子.

英文摘要：

We discuss a set-valued generalization of strong proximinality in Banach spaces, introduced by J. Mach [Continuity properties of Chebyshev centers, J. Approx. Theory, 29(3):223--230, 1980] as property-$(P_1)$. We establish that if the closed unit ball of a closed subspace of a Banach space $X$ possesses property-$(P_1)$ for each of the classes of closed bounded, compact and finite subsets of $X$, then so does the subspace. It is also proved that the closed unit ball of an $M$-ideal in an $L_{1}$-predual space satisfies property-$(P_{1})$ for the compact subsets of the space. For a Choquet simplex $K$, we provide a sufficient condition for the closed unit ball of a finite co-dimensional closed subspace of $A(K)$ to satisfy property-$(P_{1})$ for the compact subsets of $A(K)$. This condition also helps to establish the equivalence of strong proximinality of the closed unit ball of a finite co-dimensional subspace of $A(K)$ and property-$(P_{1})$ of the closed unit ball of the subspace for the compact subsets of $A(K)$. Further, for a compact Hausdorff space $S$, a characterization is provided for a strongly proximinal finite co-dimensional closed subspace of $C(S)$ in terms of property-$(P_{1})$ of the subspace and that of its closed unit ball for the compact subsets of $C(S)$. We generalize this characterization for a strongly proximinal finite co-dimensional closed subspace of an $L_{1}$-predual space. As a consequence, we prove that such a subspace is a finite intersection of hyperplanes such that the closed unit ball of each of these hyperplanes satisfy property-$(P_1)$ for the compact subsets of the $L_1$-predual space and vice versa. We conclude this article by providing an example of a closed subspace of a non-reflexive Banach space which satisfies $1 \frac{1}{2}$-ball property and does not admit restricted Chebyshev centre for a closed bounded subset of the Banach space.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文