Real Analysis

Abstract

3. Since Lp and Lr are subspaces of CX , their intersection is a vector space. It is clear that ‖ · ‖ is a norm (this follows directly from the fact that ‖ · ‖p and ‖ · ‖r are norms). Let 〈fn〉n=1 be a Cauchy sequence in Lp ∩ Lr. Since ‖fm − fn‖p ≤ ‖fm − fn‖ and ‖fm − fn‖r ≤ ‖fm − fn‖ for all m,n ∈ N, it is clear that 〈fn〉n=1 is a Cauchy sequence in both Lp and Lr. Let gp ∈ Lp and gr ∈ Lr be the respective limits of this sequence. Given ε ∈ (0,∞), there exists N ∈ N such that ‖fn − gp‖p < ε(p+1)/p for all n ∈ N with n ≥ N. If n ∈ N and n ≥ N