# Understanding Projection operator $\pi$

I know that the projection function $$\pi_{i}(x)=a_{i}$$

where $$x=(a_{1}, a_{2}, \cdots a_{n})$$ is a point in the n dimensional space defined by the cartesian product $$A_{1} \times A_{2} \times \cdots A_{n}$$ and $$a_{i}$$ is the $$i^{th}$$ coordinate of $$x$$. Source: https://solitaryroad.com/c787.html

Then what does $$\pi_{[-c,c]^{n}}(z)$$ mean?

where $$[-c, c]^{n} = [-c,c] \times\cdots [-c,c]$$ and c is always positive.

• A bit more context might help; what are the domain and codomain of this map? Can you quote an excerpt where this notation is used? – Servaes Jun 12 at 14:29

## 1 Answer

It's the projection of the point $$z$$ onto the cube $$[-c,c]\times[-c,c]\times\cdots\times[-c.c]$$. So, if $$z=(z_1, z_2, \ldots, z_n)$$ are the co-ordinates of $$z$$, for each co-ordinate you get $$z_i$$ if $$z_i\in [-c,c]$$ and $$0$$ otherwise. As an example, suppose $$n=3$$ and $$z=(5,-2,1.5)$$. Then $$\pi_{[-2,2]^3}(z) = (0,-2, 1.5)$$

Note that normally a projection is looking at an object with more dimensions that the space being projected to, but it's not necessary