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math214:02-10

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2020-02-10, Monday

$$\gdef\wt \widetilde, \gdef\RM\backslash$

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, if $F$ is already smooth on a closed set $A$, we can choose $\wt F = F$ on $A$.

Sketch of the proof: We do it following steps

  1. By extension of smooht function lemma, we may find a smooth function $F_0: M \to \R$ that agrees with $F$ on $A$. Define $$ U_0 = \{x \in M | | F(x) - F_0(x)| < \delta(x) \}.$$
  2. For each point $x \in M$, define $U_x = \{ y \in M \RM A |F(y) - F(x)| < \delta(x)/2, \z{ and } \delta(x)/2 < \delta(y) \}$.
  3. The collection of open sets $\{U_x\}$ covers $M \RM A$, we choose a countable subcover
math214/02-10.1581352803.txt.gz · Last modified: 2020/02/10 08:40 by pzhou