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math214:02-10 [2020/02/10 08:40] pzhou created |
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====== 2020-02-10, Monday ====== | ====== 2020-02-10, Monday ====== | ||
- | $$\gdef\wt \widetilde, \gdef\RM\backslash$ | + | $$\gdef\wt\widetilde \gdef\RM\backslash$$ |
===== Whitney Approximation Theorem. ===== | ===== Whitney Approximation Theorem. ===== | ||
**Thm** Suppose $M$ is a smooth manifold, $F: M \to \R^k$ is a continuous map. $\delta: M \to \R$ is a positive function. Then, we can find a smooth function $\wt F: M \to \R^k$, such that $|F(x) - \wt F(x)| < \delta(x)$ for all $x \in M$. Furthermore, | **Thm** Suppose $M$ is a smooth manifold, $F: M \to \R^k$ is a continuous map. $\delta: M \to \R$ is a positive function. Then, we can find a smooth function $\wt F: M \to \R^k$, such that $|F(x) - \wt F(x)| < \delta(x)$ for all $x \in M$. Furthermore, | ||
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- By extension of smooht function lemma, we may find a smooth function $F_0: M \to \R$ that agrees with $F$ on $A$. Define $$ U_0 = \{x \in M | | F(x) - F_0(x)| < \delta(x) \}.$$ | - By extension of smooht function lemma, we may find a smooth function $F_0: M \to \R$ that agrees with $F$ on $A$. Define $$ U_0 = \{x \in M | | F(x) - F_0(x)| < \delta(x) \}.$$ | ||
- For each point $x \in M$, define $U_x = \{ y \in M \RM A |F(y) - F(x)| < \delta(x)/ | - For each point $x \in M$, define $U_x = \{ y \in M \RM A |F(y) - F(x)| < \delta(x)/ | ||
- | - The collection of open sets $\{U_x\}$ covers $M \RM A$, we choose a countable subcover | + | - The collection of open sets $\{U_x\}$ covers $M \RM A$, we choose a countable subcover |
+ | - Do a partition of unity $\{\varphi_0, | ||
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+ | ===== Tubular Neighborhood Theorem ===== | ||
+ | The big plan: we want to be able to approximate a $C^0$ map $F: N \to M$ by a $C^\infty$ map $\wt F: N \to M$, such that the $C^0$ distance of $F$ and $\wt F$ is small. | ||
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