推 tasukuchiyan:一直卡在limsup這步，沒想到這麼簡潔漂亮，感謝您 06/11 09:55
※ 引述《tasukuchiyan (Tasuku)》之銘言：
: Let f be a complex measurable function.
: Assume 0 < ||f||_∞ < ∞ and ||f||_r < ∞ for some r < ∞.
: Prove that ||f||_p → ||f||_∞ as p → ∞.
1.let ║f║ = M
∞
║f║ ≧ [∫ |f|^p ]^(1/p) ≧ ( M-ε) |{|f|≧ M-ε}|^(1/p)
p {|f|≧ M-ε}
=> liminf ║f║ ≧ liminf ( M-ε) |{|f|≧ M-ε}|^(1/p)
p
= M-ε = ║f║ -ε
∞
2. let p > r
(p-r)/p
║f║ = [∫|f|^r |f|^(p-r) ]^(1/p) ≦ ║f║ [∫|f|^r ]^(1/p)
p ∞
(p-r)/p
=> limsup ║f║ ≦ limsup ║f║ [∫|f|^r ]^(1/p)
p ∞
= ║f║
∞
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.115.222.5