Compact sets in the space \(L^ p(0,T;B)\). (English) Zbl 0629.46031

Simon, Jacques

Ann. Mat. Pura Appl., IV. Ser. 146, 65-96 (1987).

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

MSC:

46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46A50	Compactness in topological linear spaces; angelic spaces, etc.
46E40	Spaces of vector- and operator-valued functions

Keywords:

space of continuous vector valued functions; Bochner space; uniform convergence norm; theorem of Gelfand-Phillips-Nakamura

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