Best Approximation

$W$ is a subspace of ${\mathbb{R}}^n$, $\vec{y}\in {\mathbb{R}}^n$, $\hat{y}=pro{j}_W\vec{y}$ $⟹$

$$\forall \vec{w}\ne \hat{y}\in W||\vec{y}-\hat{y}||<||\vec{y}-\vec{w}||$$

In other words, $\hat{y}$ is closest vector in $W$ to $\vec{y}$ See Orthogonal Decomposition, especially for the visual interpretation of this.