A Banach space in which all compact sets, but not all by Vesely L.

By Vesely L.

Given a Banach spaceX, letc 0(X) be the distance of all null sequences inX (equipped with the supremum norm). We convey that: 1) every one compact set inc 0(X) admits a (Chebyshev) heart iff every one compact set inX admits a middle; 2) forX gratifying a undeniable situation (Q), each one bounded set inc 0(X) admits a middle iffX is quasi uniformly rotund. We build a Banach spaceX such that the compact subsets ofX admit centers,X satisfies the situation (Q) andX isn't quasi uniformly rotund. It follows that the Banach spaceE=c 0(X) has the valuables from the identify.

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The origin and evolution of these traits raise some intriguing evolutionary questions. Many different species of animals are unpalatable, poisonous, or malodorous to potential predators. Many of these species with distasteful qualities have bright colors arranged in conspicuous patterns. Tests of palatability of cryptically and conspicuously colored members of related groups of species have revealed that predators find a much higher proportion of the cryptically colored species to be palatable.

To the extent that these phenotypes are controlled or influenced by genes or genotypes, natural selection will act indirectly on genes and genotypes. If the genotypes of individuals with different phenotypes differ, those genes and genotypes of the individuals favored by natural selection will tend to increase in relative frequency. In this way natural selection brings about evolutionary change. Because 18 ADAPTATION population geneticists are preoccupied with changes in gene frequency and with evolutionary change, they frequently refer to selection favoring a particular gene or genotype.

In winter male goldfinches molt into a subdued olivedrab plumage so that even though flocks of goldfinches may winter in the north, identification is difficult at first because the male's win- ADAPTATION 23 ter and summer plumages are so different. Darwin's work on sexual selection was an attempt to explain the adaptive significance and the origin of the differences between males and females in their secondary sexual characteristics as well as in their primary sexual characteristics. The sexes often differ not just in their external appearance but in their physiology and behavior.

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