In mathematics, Souček spaces are generalizations of Sobolev spaces, named after the Czech mathematician Jiří Souček. One of their main advantages is that they offer a way to deal with the fact that the Sobolev space W1,1 is not a reflexive space; since W1,1 is not reflexive, it is not always true that a bounded sequence has a weakly convergent subsequence, which is highly desirable in many applications.
Definition Let Ω be a bounded domain in n-dimensional Euclidean space with smooth boundary. The Souček space W1,μ(Ω; Rm) is defined to be the space of all ordered pairs (u, v), where
u lies in the Lebesgue space L1(Ω; Rm); v (thought of as the gradient of u) is a regular Borel measure on the closure of Ω; there exists a sequence of functions uk in the Sobolev space W1,1(Ω; Rm) such that
lim k → ∞ u k = u in L 1 ( Ω ; R m ) {\displaystyle \lim _{k\to \infty }u_{k}=u{\mbox{ in }}L^{1}(\Omega ;\mathbf {R} ^{m})}
and
lim k → ∞ ∇ u k = v {\displaystyle \lim _{k\to \infty }\nabla u_{k}=v}
weakly-∗ in the space of all Rm×n-valued regular Borel measures on the closure of Ω.
Properties The Souček space W1,μ(Ω; Rm) is a Banach space when equipped with the norm given by
‖ ( u , v ) ‖ := ‖ u ‖ L 1 + ‖ v ‖ M , {\displaystyle \|(u,v)\|:=\|u\|_{L^{1}}+\|v\|_{M},}
i.e. the sum of the L1 and total variation norms of the two components.
References Souček, Jiří (1972). "Spaces of functions on domain Ω, whose k-th derivatives are measures defined on Ω̅". Časopis Pěst. Mat. 97: 10–46, 94. doi:10.21136/CPM.1972.117746. ISSN 0528-2195. MR 0313798
