Let $g:[a,b] \rightarrow\mathbb R$ be continuous and strictly increasing. Show that $\{p \circ g : p \text{ is a polynomial}\}$ is dense in $C[a,b]$.
Prove that if the function $f$ be continuous and strictly increasing is dense.
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real-analysis
sequences-and-series
continuity
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2$g$ is a homeomorphism and $\{p\in C([0,1]): p \text{ is a polynomial}\}$ is dense in $C([0,1])$. – 2017-02-09
1 Answers
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Here is an outline:
Since $g$ is strictly monotone, it has a continuous inverse. Let $[c,d] = g([a,b])$ and select $f \in C[a,b]$.
Consider approximating $f \circ g^{-1}$ with a polynomial $p$ and how you use this to answer the question.
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0Can't we use Stone-Weierstrass theorem? – 2017-02-14
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0Yes, that is implicit above. Use the Weierstrass approximation theorem to find a polynomial $p$ that approximates $f \circ g^{-1}$. – 2017-02-14