A counterexample to a compact embedding theorem for functions with values in a Hilbert space
Stanisław Migórski · International Journal of Stochastic Analysis · 1995
A counterexample to a compactness embedding result of Nagy is provided. Let V and H be real separable Hilbert spaces with V densely and continuously embedded in H. Identifying H with its dual we write V c H c V' algebraically and topologically, where V' is the dual space to V. Given T > 0, let Y = L2(0, T; V), * = L2(0, T; H) and 7' = L2(0, T; V') denote the spaces of the square summable functions defined on the interval (0, T) with values in V, H and V', respectively. We define 2 = {v E 7: v' E 7'},whichwiththeusualnorm Ilullr = (I1uII2+I1u'II2,)1/2 is a Hilbert space (cf. [7], Theorem 25.4). The following remarks about the space 2f are in order. In the definition the derivative u' = du is understood in the dt weak sense. Nevertheless, when we view u as a V'-valued function, we know (see [8], Proposition 23.23) that it is absolutely continuous, so its derivative exists in the strong sense almost everywhere on (0, T). It is known (see, e.g., [7], Theorem 25.5) that the embedding (1) 7C C(0 T; H) is continuous. Here C(0, T; H) denotes the space of continuous functions from [0, T] into H, endowed with the supremum norm. Therefore every function in /l can be, after modification on the set of measure zero, considered as an element of C(0, T; H). We also know that 2 c X compactly (see [8]). The following is the main result of Nagy (see [3], Theorem 2). Theorem. Let V and H be infinite-dimensional separable Hilbert spaces such that the embedding V c H is dense, continuous and compact. Then the embedding (1) is also compact. The aim of this paper is to exhibit a simple example which shows that the embedding (1) cannot be compact, i.e., the above theorem is not true. It should also be noted here that this theorem was exploited in several recent papers in Received by the editors December 1, 1993. 1991 Mathematics Subject Classification. Primary 46E35.