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Let $C_{e}([-1,1],\mathbb{R})$ denote the set of even functions in $C([-1,1],\mathbb{R})$

(a) Show $C_e$ is closed and not dense in $C$.

(b) show the even polynomials are dense in $C_e$, but not in $C$.

I can't start on it... I can't catch any clue..

kahen
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syko
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  • What is the metric on your space? Uniform approximation? – awwalker May 16 '13 at 04:30
  • I cant understand what you mean. What does you intend by 'my space'? – syko May 16 '13 at 04:56
  • The space $C_e([-1,1],\mathbb{R})$ is a metric space under many metrics. I assume (but wanted it to be clear) that your intended metric was the sup norm. – awwalker May 16 '13 at 05:04
  • I dont know exact meaning of the sup norm. but when I leanred it in university, the norm of function is defined by its sup on domain. So I guess it is the sup norm – syko May 16 '13 at 05:08

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Assume there is a sequence of even functions $f_{i}$ converge to $x$ under $\sup$ norm such that $$\sup |f_{i}(x)-x|<\epsilon$$ for large enough $i$. Then this implies $$\int f_{i}=\int x$$ on $[-R,R]$. But this is impossible unless $\int f_{i}\rightarrow 0$ on $[0,R]$ for any $R$. However this itself is absurd. $(2)$ is similar.

Bombyx mori
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  • Thanks a lot, but how can I show it is a closed set in prob(a). Using the fact that continuous function makes an image which is compact if domain is compact? – syko May 16 '13 at 04:54
  • You only need to show the limit of even functions is even. Assume not and you can prove it without much effort by making use of continuity. – Bombyx mori May 16 '13 at 05:11