projection onto vector spaces

Question

How do you project a vector on to the euclidean ball? For example, if there is a vector $x ∈ R^n$ how does one project this onto the euclidean ball.

What are the steps for projecting a vector onto a subspace? Is there a formula?

Answer 1

The usual thing would be to divide $x$ by its magnitude, assuming that $x \ne 0$. Then $$\frac{x}{\|x\|}$$ would be a vector whose magnitude is exactly $1$, and so can be thought of as a point on the surface of the ball.