Inner product/Gradient/Restriction/Orthogonal projection/Exercise

Let ${}V$ be a Euclidean vector space, and ${}U\subseteq V$ a linear subspace, endowed with the induced inner product. Let

$f\colon V\longrightarrow \mathbb {R}$

be a linear form, and let ${}v\in V$ denote the corresponding gradient in the sense of fact. Show that the gradient ${}u\in U$ of the restriction ${}f{|}_{U}$ equals the orthogonal projection of ${}v$ to ${}U$.

Create a solution