Examples discussion 04
نویسنده
چکیده
[04.1] Comparing L spaces. Let 1 ≤ p, p′ < ∞. When is L[a, b] ⊂ Lp [a, b] for finite intervals [a, b] and Lebesgue measure? When is L(R) ⊂ Lp(R)? When is ` ⊂ `p? Discussion: Take p < p′. We claim that L[a, b] ⊃ Lp [a, b], with proper containment. The function f that is (x − a) 1 p′ on (a, b] and 0 off that interval is not in L ′ , but is in L. Given f ∈ Lp [a, b], let E be the set of x ∈ [a, b] where |f(x)| ≥ 1. Then ∫ b a |f |p < ∞ if and only if ∫ E |f |p < ∞. On E, |f | < |f |p , so ∫ E |f | <∞, and then also ∫ b a |f | <∞, so f ∈ L[a, b]. ///
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تاریخ انتشار 2016