Suppose there exists a sequence [itex]f_n[/itex] of square-integrable functions on [itex]\mathbb R[/itex] such that [itex]f_n(x) \to f(x)[/itex] in the L^2-norm with [itex]x \ f_n(x) \to g(x)[/itex], also in the L^2-norm. We know from basic measure theory that there's a subsequence [itex]f_{n_k}[/itex] with [itex]f_{n_k}(x) \to f(x)[/itex] for a.e. x. But my professor seems to be claiming that this somehow implies [itex]x \ f_{n_k}(x) \to g(x)[/itex] for a.e. x. I don't see why this is. Obviously, we know that [itex]x \ f_{m_k} \to g[/itex] a.e. for SOME subsequence of [itex]f_n[/itex]...but how do we know it works for the SAME subsequence? Can someone offer some guidance? Thanks!(adsbygoogle = window.adsbygoogle || []).push({});

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# Convergence in the L^2 norm

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