Integration in Locally Compact Spaces II
Integration in Locally Compact Spaces II
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局部紧凑空间的集成 II
DOI：
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发表时间：
1951
影响因子：
0.8
通讯作者：
H. Zuckerman
中科院分区：
文献类型：
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作者：
E. Hewitt;H. Zuckerman
赋范线性空间上连续线性泛函的具体表示的一般问题，即，的识别共轭空间，当然吸引了许多数学家的注意，在过去的五十年里，并已解决了许多情况下[1，页。59-72]。同样地，扩展定义在赋范线性空间的线性子空间上的线性泛函的问题可以被认为是由哈恩-巴拿赫定理[1，第28页]解决的，尽管涉及“自然”扩展的问题，比如从黎曼积分得到勒贝格积分的问题，仍然存在。在本文中，我们将考虑两个“自然”的方法，延长一定的线性功能，并表明他们实际上是相同的。作为一个副产品，我们得到一个具体的表示为原来的功能和其“自然”的扩展。在随后的通信中，作者将考虑拓扑在某些家庭的线性泛函，规范决议的线性泛函，和其他扩展问题。
The general problem of producing concrete representations for continuous linear functionals on normed linear spaces, ie., of identifying conjugate spaces, has of course attracted the attention of many mathematicians during the last five decades and has been solved in many cases [1, pp. 59-72]. Likewise, the problem of extending a linear functional defined on a linear subspace of a normed linear space may be regarded as solved by the Hahn-Banach theorem [1, p. 28], although problems involving “natural” extensions, like that yielding the Lebesgue integral from the Riemann integral, remain. In the present paper, we shall consider two “natural’ methods of extending a certain linear functional and show that they are in fact identical. As a by-product, we obtain a concrete representation both for the original functional and for its “natural” extension. In subsequent communications, the writers will consider topologies in certain families of linear functionals, canonical resolutions of linear functionals, and other extension problems.