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functional analysis Haim brezis 4 . 5 this all the questions there is not any thing 1 . Prove that L ^ 1 (

functional analysis Haim brezis 4.5 this all the questions there is not any thing 1. Prove that L^1(\Omega )\cap L^\infty (\Omega ) is a dense subset of L^p(\Omega ).2. Prove that the set {f in L^p(\Omega )\cap L^q(\Omega ) ;f_q<=1} is closed in L^p(\Omega ).3. Let (f_n) be a sequence in L^p(\Omega )\cap L^q(\Omega ) and let f in L^p(\Omega ). Assume that f_n-> f in L^p(\Omega ) and f_n_q<= C . Prove that f in L^r(\Omega ) and that f_n-> f in L^r(\Omega ) for every r between p and q, r != q.

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