7. The Whitney Theorems

In this lecture we prove two celebrated theorems of Whitney:

• The Whitney Embedding Theorem, which states any smooth manifold of dimension n may be embedded in $ \mathbb{R}^{2m}$ (we only prove a weaker version: namely that any compact smooth manifold of dimension $m$ may be embedded in $ \mathbb{R}^{2m+1}$).
• The Whitney Approximation Theorem, which states that a continuous map $ h \colon M \to N$ is homotopic to a smooth map $ \varphi \colon M \to N$. We will use this at the end of the semester to establish the homotopy invariance property of de Rham cohomology.