How to solve the following problem:Let f n , f ∈ L 2 ( R d ) for all n ≥ 1 be such that ‖ f n ‖ 2 → ‖ f ‖ 2 as n → ∞. Suppose, moreover, that ∫ f n g → ∫ f gfor all g ∈ L 2 ( R d ). Then f n converges to f in L 2 -norm.

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How to solve the following problem:Let       f    n    ,  f  ∈      L    2    (            R        d    ) for all   n  ≥  1 be such that   ‖      f    n        ‖    2    →  ‖  f      ‖    2   as   n  →  ∞. Suppose, moreover, that  ∫      f          n        g  →  ∫  f  gfor all   g  ∈      L    2    (            R        d    ). Then       f    n   converges to   f in       L    2  -norm.

Oliver Shepherd · Accepted Answer

You want to show  ‖  f  −      f    n        ‖    2    →  0Equivalently, you can show  ∫  (  f  −      f    n        )    2    d  μ  =  ‖  f  −      f    n        ‖    2    2    →  0We have  ‖  f  −      f    n        ‖    2    2    =  ∫  (  f  −      f    n        )    2    d  μ  =  ∫      f    2    d  μ  −  2  ∫  f      f    n    d  μ  +  ∫      f    n    2    d  μSince we have   ‖      f    n        ‖    2    →  ‖  f      ‖    2  , we have   ∫      f    n    2    d  μ  →  ∫      f    2    d  μ and since we have   ∫      f    n    g  d  μ  →  ∫  f  g  d  μ we have   ∫  f      f    n    d  μ  →  ∫      f    2    d  μ so that  ∫      f    2    d  μ  −  2  ∫  f      f    n    d  μ  +  ∫      f    n    2    d  μ  →  ∫      f    2    d  μ  −  2  ∫      f    2    d  μ  +  ∫      f    2    d  μ  =  0that is,  ‖  f  −      f    n        ‖    2    2    →  0and hence of course also  ‖  f  −      f    n        ‖    2    →  0

bandikizaui · Accepted Answer

Since       L    2   is a Hilbert space, you can use the parallelogram identity. More generally, you can also use a property of any uniformly convex Banach space. A very nice proof appears in Brezis&#039; book on functional analysis.

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