math214:02-10
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math214:02-10 [2020/02/10 09:05] pzhou [Whitney Approximation Theorem.] |
math214:02-10 [2020/02/11 20:44] pzhou |
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| ====== 2020-02-10, Monday ====== | ====== 2020-02-10, Monday ====== | ||
| - | $$\gdef\wt \widetilde, \gdef\RM\backslash$ | + | $$\gdef\wt\widetilde \gdef\RM\backslash$$ |
| ===== Whitney Approximation Theorem. ===== | ===== Whitney Approximation Theorem. ===== | ||
| **Thm** Suppose $M$ is a smooth manifold, $F: M \to \R^k$ is a continuous map. $\delta: M \to \R$ is a positive function. Then, we can find a smooth function $\wt F: M \to \R^k$, such that $|F(x) - \wt F(x)| < \delta(x)$ for all $x \in M$. Furthermore, | **Thm** Suppose $M$ is a smooth manifold, $F: M \to \R^k$ is a continuous map. $\delta: M \to \R$ is a positive function. Then, we can find a smooth function $\wt F: M \to \R^k$, such that $|F(x) - \wt F(x)| < \delta(x)$ for all $x \in M$. Furthermore, | ||
math214/02-10.txt · Last modified: 2020/02/11 20:47 by pzhou
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