Let (X, A, ) be a measure space. Let p, q belongs to [1, ] be...

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Let (X, A, μ) be a measure space. Let p, q belongs to [1, ∞] be conjugate. Let (fn)neN C LP(X) and f LP(X) and similarly (gn)neN C L'(X) and g E L'(X). Show that lim ||fn - fl|p=0_and_lim ||gn - 9||q=0] →lim || fn9n - fg||1 = 0. Ln-∞0 1-∞ Let (X, A, μ) be a measure space. Let p, q belongs to [1, ∞] be conjugate. Let (fn)neN C LP(X) and f LP(X) and similarly (gn)neN C L'(X) and g E L'(X). Show that lim ||fn - fl|p=0_and_lim ||gn - 9||q=0] →lim || fn9n - fg||1 = 0. Ln-∞0 1-∞

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