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--- math214:02-10 [2020/02/10 09:05]
pzhou [Whitney Approximation Theorem.]
+++ math214:02-10 [2020/02/11 20:47] (current)
pzhou [Tubular Neighborhood Theorem]
@@ Line 1: / Line 1: @@
 ====== 2020-02-10, Monday ======
-$$\gdef\wt \widetilde, \gdef\RM\backslash$
+$$\gdef\wt\widetilde \gdef\RM\backslash$$
 ===== Whitney Approximation Theorem. =====
 **Thm** Suppose $M$ is a smooth manifold, $F: M \to \R^k$ is a continuous map. $\delta: M \to \R$ is a positive function. Then, we can find a smooth function $\wt F:  M \to \R^k$, such that $|F(x) - \wt F(x)| < \delta(x)$ for all $x \in M$. Furthermore, if $F$ is already smooth on a closed set $A$, we can choose $\wt F = F$ on $A$.
@@ Line 11: / Line 12: @@
 ===== Tubular Neighborhood Theorem =====
-The big plan: we want to be able to approximate a $C^0$ map $F: N \to M$ by a $C^\infty$ map $\wt F: N \to M$, such that the $C^0$ distance of $F$ and $\wt F$ is small.
+The big plan: we want to be able to approximate a $C^0$ map $F: N \to M$ by a $C^\infty$ map $\wt F: N \to M$, such that the $C^0$ distance of $F$ and $\wt F$ is small. In order to do this, we first embed $M$ to $\R^m$ for some big $m$,
+$$ \iota: M \into \R^m $$
+ then we smooth the composition $\iota \circ F: N \to M$, to get $\wt G: N \to \R^m$, $C^0$-close to the original image of $\iota(M)$. Finally, we project $\wt G(N)$ back onto $M$. This smoothing-then-project-back operation gives a smooth map from $N$ to $M$.

Lecture Notes