→ ppia :Note ║f║_n→║f║_∞ since μK < ∞. 07/23 10:46
推 kemowu :thanks a lot!! 07/23 16:50
※ 引述《kemowu (小展)》之銘言：
: Let (X, M, μ) be a measure space with μ(X)<∞.
: ∞ n
: Suppose f is in L (μ), define α_n = ∫ |f| dμ.
: X
: α_n+1
: Show that lim －－－－ = ║f║ .
: n→∞ α_n ∞
n+1 n
Indeed, |f| ≦ |f|║f║_∞ a.e..Hence α_{n+1}/α_n ≦1.
For the reversed inequality, utilize Holder inequality to obtain
n ╭ ╮1/(n+1)╭ n+1 ╮n/(n+1)
∫ |f| dμ ≦ │∫1 dμ│ │∫ |f| dμ│.
X ╰ X ╯ ╰ ╯
1/(n+1) -1 -1
Hence α_n/α_{n+1} ≦ (μK) ║f║ → ║f║ .
n ∞
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◆ From: 114.32.4.99
※ 編輯: ppia 來自: 114.32.4.99 (07/22 20:26)