Is $C^\infty (D^n,S^n)$ compact?

Question

Is $ C^\infty (D^n,S^n)$ compact where $D^n$ is the unit closed disc in $\mathbb{R^n}$? I got this question when reading a paper. The author mentioned without explanation that $\{ \alpha \in \Gamma (\bar{U} , E) \big| \lVert \alpha \rVert_x = 1 \ \textrm{for all} \ x \in U \}$ is compact where $E$ is a Hermitian vector bundle over a Riemannian manifold $M$ and $U$ is a local chart of it.

edited:

The author considered an inequality, $$ \inf_{x \in \bar{U}} \lVert \sum_{i=1}^n \sigma_j Z_j \alpha \rVert_x^2 \geq c^2 $$ where $c \in \mathbb{R}$ is a constant, $Z_j$'s are some bundle maps, $\sigma = (\sigma_j)$ is taken over $S^{n-1}$, and $\alpha$ is taken over the set mentioned above. He claims that $S^{n-1} \times \{ \alpha \in \Gamma (\bar{U} , E) \big| \lVert \alpha \rVert_x = 1 \ \textrm{for all} \ x \in U \}$ is compact so the left hand side has minimum and that will be greater than $c^2$.

Answer 1

Note that $$\lVert \sum_{i=1}^n \sigma_j Z_j \alpha \rVert_x^2$$ depends on $\alpha$ only through $\alpha(x)$. Since any $\beta \in E_x$ with $x\in \bar U$, $\Vert \beta \Vert = 1$ can be extended to an admissible $\alpha$ with $\alpha(x) = \beta$, your minimization over some sections reduces to one over some of the total space:

$$ \inf_{\alpha \in \Gamma(\bar U, E), \Vert \alpha \Vert = 1} \inf_{x \in \bar{U}} \lVert \sum_{i=1}^n \sigma_j Z_j \alpha \rVert_x^2 \geq c^2 = \inf_{\beta \in S(E)|_\bar U} \lVert \sum_{i=1}^n \sigma_j Z_j \beta \rVert^2$$

where $$S(E)|_\bar U= \{ \beta \in E : \pi(\beta) \in \bar U, \Vert \beta \Vert = 1 \}.$$

Assuming $U$ is bounded we see that $S(E)|_\bar U$ is compact, so the minimum is attained for some $\beta$. Thus any extension of this $\beta$ to a section $\alpha$ is a minimizer for your original problem.

Getting topology of function spaces involved seems completely unnecessary to me, and (at least for a simple continuity-compactness argument) doesn't look straightforward in this case - even in the $L^\infty = C^0$ norm you need to impose some kind of equicontinuity/gradient bound condition to get compactness.