On the volume of unit balls in Banach spaces
On the volume of unit balls in Banach spaces
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关于 Banach 空间中单位球的体积
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发表时间：
1982
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通讯作者：
C. Schütt
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作者：
C. Schütt
我们估计了lpn ~ nlrn、1 ~ p、r ~ ~、酉算子理想和对称空间的体积比。我们还研究了n维James空间的结构。本文研究了有限维Banach空间中单位球的体积和这种空间的一个不变量体积比。我们首先给出单位球体积的估计。特别是，对于具有1-无条件基的空间，我们得到简单的公式。这些公式以一种自然的方式推广到没有1-无条件基的空间。然后，我们估计了文献[11]中提出的1-对称空间和lpn ~ 03 C 0 lrn，1 p，r ~ 00的体积比.我们用一种简单的方法得到了n维James空间Jn的体积比的估计。我们还证明了Jn和1的Banach-Mazur距离至多为log n阶，并且Jn [8]的k-常数是一致有界的.关于单位球体积的其他方面在[12]中考虑。0.在本文中，我们估计的单位球的体积空间Rn提供各种标准。这是通常的勒贝格测度。因为我们总是考虑Rn，所以我们对两个空间之间的自然恒等式的理解也是清楚的。我们用BE表示Banach空间E的单位球，用Sn-1表示1中的单位球。空间E的体积比由0010- 437 X/82090393-15$00.20/0394给出，其中e是椭球。用Xn表示1中单位球的体积。定义了两个Banach空间E和F的Banach-Mazur距离，其中一个是同构的，另一个是由对称空间E诱导的酉算子理想。如果E = 1 P，我们写Cp。8-张量积是具有最小张量范数的张量积，而n-张量积是具有最大张量范数的张量积。我们说{ei}ni=1是E的C-无条件基，如果对所有ai E R，ei = + 1，i = 1，.，n，并且它是C对称的，如果对于所有a; E R，gi = ± 1，i = 1，.，n和（1，.、n）的函数。对偶基由{e*i}ni=1表示。如果索引太笨拙，我们写e（i）而不是ei。n维詹姆斯空间是Rn，其范数为sup覆盖所有严格增序列。欧几里得单位球的体积是395 1。单位球体积的基本估计我们在这里介绍单位球体积的估计。我们从E有无条件基的情况开始。可以根据某些矢量来估计体积。这些向量的存在性已在[4]、[7]中得到了证明。引理1.1：设{ei}ni=1是E的1-无条件基.则存在一个真实的数序列{si}ni=1，使得引理1.2：设{ei}ni=1是E的一个1-无条件基。对于每个序列si，ti，i = 1，.，n使得siti = 1 /n，并且
We estimate the volume ratio of lpn ~n lrn, 1 ~ p, r ~ ~, unitary operator ideals and symmetric spaces. We also study the structure of the n-dimensional James pace. We consider the volume of unit balls in finite dimensional Banach spaces and an invariant of such spaces, the volume ratio. We start with giving estimations for the volume of unit balls. In particular, for spaces with 1-unconditional bases we get simple formulas. These formulas are extended in a natural way to spaces without 1-unconditional bases. Then, we estimate the volume ratio of 1-symmetric spaces and of lpn ~03C0 lrn, 1 p, r ~ oo, a problem posed in [11]. We get in an easy way estimations for the volume ratio of the n-dimensional James space Jn. We also show that the Banach-Mazur distance of Jn and 1; is at most of the order log n and that the ké-constant of Jn [8] is uniformly bounded. Other aspects concerning volumes of unit balls are considered in [12]. 0. Preliminaries In this paper we estimate volumes of unit balls of the space Rn provided with various norms. The measure is the usual Lebesgue measure. Since we always consider the Rn it is also clear what we understand by the natural identity between two spaces. We denote the unit ball of a Banach space E by BE and the unit sphere in 1; by Sn-1. The volume ratio of a space E is given by 0010-437X/82090393-15$00.20/0 394 where e is an ellipsoid. By Xn we denote the volumes of the unit balls in 1;. The Banach-Mazur distance of two Banach spaces E and F is defined by is isomorphism} One has By CE we understand the unitary operator ideal with the norm induced by the symmetric space E. If E = 1P we write Cp. The 8-tensor product is the tensor product with the smallest tensor norm and the n-tensor product that with the biggest tensor norm. We say that {ei}ni=1 is a C-unconditional basis of E if for all ai E R, ei = + 1, i = 1,..., n, and that it is C-symmetric if for all a; E R, gi = ± 1, i = 1,..., n and all permutations n of ( 1 , ... , n) . The dual basis is denoted by {e*i}ni=1. If the indices are too awkward we write e(i) instead of ei. The n-dimensional James space is Rn with the norm where the sup is taken over all strictly increasing sequences. The volume of the Euclidean unit ball is 395 1. Basic estimâtes for volumes of unit balls We introduce here estimates for the volumes of unit balls. We start with the case of E having an unconditional basis. It is possible to estimate the volume in terms of certain vectors. The existence of these vectors were proved in [4], [7]. LEMMA 1.1: Let {ei}ni=1 be an 1-unconditional basis of E. Then there is a sequence {si}ni=1 of real numbers such that LEMMA 1.2: Let {ei}ni=1 be an 1-unconditional basis of E. For every sequence si, ti, i = 1,..., n such that siti = 1 /n and