What is the Inverse of Projection $\pi$ ($proj$)?

Question

$x\in X=\Pi_i X_i$.

$proj_jx=x_j.$

Define: $proj^{-1}x_j=(0,0,...,x_j,...,0,...)$. That is, the $j$th term is $x_j$.

What is the formal name for $proj^{-1}$? I think I learned this before. It is either called something like "identity map" or "ideo" something.

Answer 1

Note that this is not really an inverse of the projection function, because there are many other vectors that project to the same number.

I don't think it has a fancy name; I would just describe it as scalar multiplication with $\mathbf e_i$.

Answer 2

The latter function is commonly called a projection. In particular, it is a projection of $x$ onto (the subspace spanned by) the standard basis vector $e_j$. If you need to distinguish them, I’ve seen $proj_j$ called a “scalar projection” or “coordinate projection,” and the other a “vector projection.”

Answer 3

There is an inverse to the projection function. The projection function is surjective and so it has a right inverse. And as it turns out, a right inverse of a projection function is again a projection function. The formal name of your inverse projection function is "a right inverse of the projection function."

To show that it is a right inverse, observe that $\text{proj}_i(\text{proj}_i^{-1}(x)) = x_i$.

Answer 4

This is natural embedding into the product; these embeddings exist when your category has a zero object, which is the case in Linear Algebra. Usually, $ι$ (greek letter iota) or $i$ is used as the symbol. In this case, it would likely be $ι_j$, i.e. the $j$th coordinate embedding.

One should avoid using $proj^{-1}$, because that kind of notation is usually reserved for functional (two-sided) inverses when they exist.