Thread: continuous function betweem two metric spaces

Hi can some please help me with the following problem:



Thanks

Hint:

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

Regards.

Fernando Revilla

Originally Posted by 1234567

Hi can some please help me with the following problem:



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.

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

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