If B denotes a Banach space and if \(T>0\), then C(0,T;B) denotes the Banach space of all continuous functions from [0,T] into B equipped with the uniform convergence norm, and for all \(1\leq p\leq \infty\), \(L^ p(0,T;B)\) is the completion of C(0,T;B) under the norm \(\| f\|_{L^ p(0,T;B)}:=(\int^{T}_{0}\| f(t)\|^ p_ Bdt)^{1/p}\) \((1\leq p<\infty)\) and \(=\| \| f(\cdot)\|_ B\|_{\infty}\) for \(p=\infty.\)
In this paper, the author reviews a number of criteria for a subset F of \(L^ p(0,T;B)\) to be compact.
The criteria presented are in a sense all reformulations of the classical theorem of Gelfand-Phillips-Nakamura stating that a bounded subset of a Banach space is relatively compact if and only if every uniformly bounded net of compact operators that converges pointwise to the identity operator converges uniformly on that subset. In addition, the paper contains a number of sufficient conditions of which the following is typical: If the sequence \(\{f_ n\); \(n=1,2,...\}\) is bounded in \(L^ q(0,T;B)\) and in \(L^ 1_{loc}(0,T;X)\), where \(X\subset B\) and if \(\{\partial f_ n/\partial t\); \(n=1,2,...\}\) is bounded in \(L^ 1_{loc}(0,T;Y)\), where \(B\subset Y\), then for all \(1\leq p<q\), \(\{f_ n\}\) is relatively compact in \(L^ p(0,T;B)\).

Reviewer: W.A.J.Luxemburg

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.