continuous function betweem two metric spaces

**1234567** · November 18th 2010, 03:53 AM

Hi can some please help me with the following problem:

continuous function betweem two metric spaces-untitled.jpg

Thanks

**FernandoRevilla** · November 18th 2010, 05:02 AM

Hint:

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

Regards.

Fernando Revilla

**Jose27** · November 18th 2010, 11:26 AM

Originally Posted by 1234567

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.

**Drexel28** · November 18th 2010, 03:00 PM

Personally while the other two responses are correct I think it's easier to notice that if $\|f-g\|_{\infty}<\frac{\varepsilon}{4}$ then $\displaystyle \left|f(1)-g(1)+(f(0)-g(0))-(2f(\frac{1}{2})-g(\frac{1}{2}))\right|\leqslant |f(1)-g(1)|+|f(0)-g(0)|+2|f(\frac{1}{2})-g(\frac{1}{2})|\leqslant 4\|f-g\|_{\infty}<\varepsilon$ .

**1234567** · November 19th 2010, 01:33 PM

Yes the method that Drexel28 is far the easy way to show this, thanks

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