ON BOUNDED BILINEAR FORMS IN AN INFINITE NUMBER OF VARIABLES
ON BOUNDED BILINEAR FORMS IN AN INFINITE NUMBER OF VARIABLES
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关于无限个变量中的有界双线性形式
DOI：
10.1093/qmath/os-1.1.164
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发表时间：
1975
影响因子：
0.7
通讯作者：
J. Littlewood
中科院分区：
文献类型：
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作者：
J. Littlewood
I （iii）我们可以区分出我们的定理的三种形式，分别是(1)amn, xm， yn是复数，(2)amn, xn， yn是实数，(3)amn是实数，xm， yn都局限于+ 1和1的值。在第（二）项说明之后，我们很容易看出这三种形式在本质上是等价的，除非另有相反的说明，否则我们在大多数情况下只局限于第（一）项。（四）我们的结果，如在定理一中那样，用级数B、C、D的“收敛性”来说明，是最容易理解的。但是，当我们考虑截断形式和级数时，当m> N或N > N时，xm， yn为零（或者，或者，amn为零），我们证明，例如，一方面，如果I< wk#在8中，那么£2\am*\*< AH>* &&；另一方面，m, N < N/存在形式QNN（具有任意大的N）使得\m， N < N在$中。{•事实上，直到最后一刻，我们所关心的正是这种结果。(v)这是由熟悉的同一性引起的
I (iii) We may distinguish three forms of our theorem in which respectively (1) the amn, xm, yn are complex,(2) the amn, xn, yn are real,(3) the amn are real and xm, yn are each restricted to the values+ 1 and—1. After remark (ii) it is easy to see that the three forms are essentially equivalent, and we shall confine ourselves for the most part, and unless the contrary is stated, to case (1).(iv) Our results are most easily understood when stated, as in Theorem 1, in terms of the'convergence'of the series B, C, D; but their true character is best brought out when we consider truncated forms and series in which xm, yn, are zero (or, alternatively, amn is zero) when m> N or n> N. We prove, eg, on the one hand that if I< WK# in 8, then£ 2\am*\*< AH>* && on the other that m, n< N/there exist forms QNN (with arbitrarily large N) such that\m, n< N in $."{• It is in fact this type of result that we shall be concerned with up to the very last moment.(v) It follows from the familiar identity