... Lp(RN) and ε>0 there exist a functiong ∈ L∞(RN) and a compact set K in RNsuch that g = 0 outside K and (11) f − gp<ε.Indeed, let χnbe the characteristic function of B(0,n) and ... into R.We shall often writef instead offdμ, and we shall also use the notation89H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, DOI 10.1007/978-0-387-70914-7_4, ... 4.4.Theorem 4.7. Lpis a vector space and pis a norm for any p, 1 ≤ p ≤∞.Proof. The cases p = 1 and p =∞are clear. Therefore we assume 1 <p<∞ and let f, g ∈ Lp.Wehave|f(x)+ g(x)|p≤...