for smooth functions ƒ and g on Ω, coinciding with some prescribed boundary function on ∂ ∂ Ω Ω , where D f ( x ) is the gradient vector at x ∈ ∈ Ω Ω . Then take the Hilbert space closure with respect to this inner product, this is the Sobolev space H 1 ( Ω Ω ) .