What is a subspace and what is a shared subspace?
I don't know what you mean by "In multi label classification we have to share the subspace.", but the concept of subspaces is fairly easy to explain.


  In mathematics, a space is a set (sometimes called a universe) with some added structure.


Source: Wikipedia

The most common spaces are $\mathbb{R}$ (the number line), $\mathbb{R}^2$ (the plane) and $\mathbb{R}^3$ (the space we often think in, with $x,y,z$ axes). Lets assume our main space is $\mathbb{R}^3$. Our data are datapoints in this space. However, we have to write a report. This report has to be printed on paper. Hence we project the data points from a 3-dimensional space to a two-dimensional subspace. So a subspace is just a part of the original space.