Let (X,A,μ) be any measure space and let u∈⋂p∈[1,∞]Lp(μ). Then limp→∞‖u‖p=‖u‖∞.
I have already proved the inequality ‖u‖∞≤lim infp→∞‖u‖p, so I still need to prove lim supp→∞‖u‖p≤‖u‖∞.
Following a hint, I have proved that for any sequence qn→∞ one has ‖u‖p+qn≤‖u‖qn/(p+qn)∞⋅‖u‖p/(p+qn)p.
How can we conclude from this that lim supp→∞‖u‖p≤‖u‖∞?
No comments:
Post a Comment