Intuition of projection of a vector on to a vector space

Question

Projection of a vector $v$ onto a subspace $K$ is given such that finding $w$

$P_K(v)$ = arg min$_{w\in K} \Vert v-w \Vert $.

What is the intuition of this expression?

Answer 1

If you think $\|u-v\|$ as a way to measure the distance between points in your space, the formula you wrote means that you define the projection as the point that is closer to $v$ among all the points $w$ in the set $K$. The reason to call such a point (if such a point exists at all and if it is the only one that has this property) "the projection of $v$ onto $K$" comes from the theory of Hilbert spaces, where if $K$ is a closed subspace, this point exists, it is unique, and it turns to be the orthogonal projection of $v$ onto the subspace $K$.