Terminology for orthogonal projections Let $H = X \oplus Y$ a Hilbert space. Then, the map $p(x + y) = x$ is called the orthogonal projection onto $X$  along $Y$.
Why is it necessary to mention along $Y$? Of course if a space has a decomposition into two subspaces $U,V$ with $U \cap V = \{0\}$ then the projection onto one of them is unique and there is no information gained by saying "along" such and such space.
 A: If you have $H = X\oplus Y$, then the projection onto $X$ is not well-defined without $Y$. The reason for this is that the splitting $H = X\oplus Y$ is not unique. That said, given $X$ there are many spaces $Y$ such that $H=X\oplus Y$.
For $H=\mathbb R^2$, $X=span\pmatrix{1\\0}$, $Y_t=span\pmatrix{t \\1}$, we have $H=X\oplus Y_t$ for all $t\in\mathbb R$.
The splitting of $z:=\pmatrix{1\\1}$ in vectors from $X$ and $Y_t$ is
$$
\pmatrix{1\\1} = (1-t)\pmatrix{1\\0} + \pmatrix{t\\1}.
$$
Here, the projection of $z$ onto $X$ along $Y_t$ is given by $(1-t)\pmatrix{1\\0}$, which clearly depends on $t$ and ultimatively on the choice of the subspace that is mentioned by 'along'.
A: The reason you want to specify "along $Y$" is that you may have to deal with multiple orthogonal projections at the same time, and then it is useful to have a way to speak of different projections without too much fuss. For example, this occurs in the Gram-Schmidt orthonormalization process, where you have to iteratively take a projection along a one-dimensional subspace and its orthogonal complement at every step. It's purely a matter of using the most convenient notation for all cases.
