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Math Help - Rapidly decreasing functions and convergence

  1. #1
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    Rapidly decreasing functions and convergence

    I am trying to show that, for f a rapidly decreasing function, the mapping from f to f' is continuous. Since the space of RDF's is an F-space, Rudin says that I can use the closed graph theorem for F-spaces, and hence I need to show that the graph of the mapping is closed.

    So I wish to show that \{(f, f') : f \in RDF's\} is closed. I think that I can do this if it follows that if f_k \to f, then f_k' \to f'?

    Can someone tell me if this is true, and if so, how would I prove it?
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  2. #2
    Super Member girdav's Avatar
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    Yes, that what you have to show. What is the topology over the space RDF?
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  3. #3
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    Quote Originally Posted by measureman View Post
    I am trying to show that, for f a rapidly decreasing function, the mapping from f to f' is continuous. Since the space of RDF's is an F-space, Rudin says that I can use the closed graph theorem for F-spaces, and hence I need to show that the graph of the mapping is closed.

    So I wish to show that \{(f, f') : f \in RDF's\} is closed. I think that I can do this if it follows that if \color{red}f_k \to f, then \color{red}f_k' \to f'?

    Can someone tell me if this is true, and if so, how would I prove it?
    The condition in red looks suspiciously like a statement of what you are trying to prove.

    I'm guessing that the F-space topology is the topology of uniform convergence on compact sets for the function and all its derivatives. In that case, the continuity of differentiation follows directly from the definition. If f_k and each of its derivatives converges uniformly to the corresponding derivative of f, then it's immediately obvious that the same is true for f_k' and f'.
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