Sobolev extension by linear operators

Let Lm,p(Rⁿ) be the Sobolev space of functions with mᵗʰ derivatives lying in Lᵖ(Rⁿ). Assume that n< p < ∞. For E ⊂ Rⁿ, let Lm,p(E) denote the space of restrictions to E of functions in Lm,p(Rⁿ). We show that there exists a bounded linear map T : Lm,p(E) → Lm,p(Rⁿ) such that, for any f ∈ Lm,p(E), we have $Tf = f$ on E. We also give a formula for the order of magnitude of \|f\|_Lm,p(E) for a given f : E → R when E is finite.