Smoothness of the projection operator

Question

Let $V$ be a smooth vector field defined on $\mathbb{R}^n$ and let $\Pi$ be a smooth $k$-dimensional subspace. For each $x \in \Pi$ consider the projection of $V(x)$ onto $\Pi$. Call this new vector field defined on $\Pi$, $\tilde{V}$. Is $\tilde{V}$ a smooth vector field?

Answer 1

When you say projection you probably mean orthogonal projection with respect to the standard euclidean inner product. Indeed the projection map $\mathbb{R}^n\rightarrow\Pi$ is smooth. One way of convincing yourself is to note that linear transformations are smooth, and that the projection map is linear.

Since the projection map is smooth, the projection of a smooth vector field is smooth because it is obtained by composing with the projection map.