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How to prove $\langle x^{2n+1}: n\in \mathbb{N}\rangle$ is dense in $\{ f\in C([0,1]): f(0)=0\}$
I'm trying to prove that $\langle x^{2n+1}: n\in \mathbb{N}_0\rangle$ is dense in $\{ f\in C([0,1]): f(0)=0\}$ without the use of the Müntz–Szász theorem.
I know how to prove this for even exponents ...
2
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1
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Is a real closed, bounded interval a locally compact Hausdorff space?
Does this hold? I've been confused by the statement of the Riesz-Markov-Kakutani representation theorem; that is, the formulation is as follows:
Let $X$ be a locally compact Hausdorff space. For ...
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behavior of function between two bounds
Let $f, U, L : [0,1] \rightarrow \mathbb{R}$ be three functions with the property that
(1) U and L are continuous functions
(2) $\forall x \in [0,1]$, $L(x) \leq f(x) \leq U(x)$
(3) $f(0)=L(0)=U(0)=...
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