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Math Help - continuous function betweem two metric spaces

  1. #1
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    continuous function betweem two metric spaces

    Hi can some please help me with the following problem:

    continuous function betweem two metric spaces-untitled.jpg

    Thanks
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Hint:

    Prove: (f_n)\rightarrow f \Rightarrow (\Delta f_n)\rightarrow \Delta f.

    Regards.

    Fernando Revilla
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  3. #3
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    Quote Originally Posted by 1234567 View Post
    Hi can some please help me with the following problem:

    Click image for larger version. 

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    Thanks
    Another thing you could do is to prove that evaluation at a point is a continous linear functional (for this it's enough to prove that the kernel of the mapping is closed, which is easy) and note that the original mapping is just a sum of these.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Personally while the other two responses are correct I think it's easier to notice that if \|f-g\|_{\infty}<\frac{\varepsilon}{4} then \displaystyle \left|f(1)-g(1)+(f(0)-g(0))-(2f(\frac{1}{2})-g(\frac{1}{2}))\right|\leqslant |f(1)-g(1)|+|f(0)-g(0)|+2|f(\frac{1}{2})-g(\frac{1}{2})|\leqslant 4\|f-g\|_{\infty}<\varepsilon.
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  5. #5
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    Yes the method that Drexel28 is far the easy way to show this, thanks
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