Corresponding a vector subspace to a point of the space.

Question

In this article http://en.wikipedia.org/wiki/Tautological_bundle we read:

Grassmannians by definition are the parameter spaces for linear subspaces, of a given dimension, in a given vector space $W$. If $G$ is a Grassmannian, and $V_g$ is the subspace of $W$ corresponding to $g$ in $G$, this is already almost the data required for a vector bundle: namely a vector space for each point $g$, varying continuously.

I don't understand the sentence " $V_g$ is the subspace of $W$ corresponding to $g$ in $G$". Indeed I don't understand the notion of corresponding a particular subspace $V_g$ to a point $g$ using the word "The subspace $V_g$ of $W$... This means there is a unique particular subspace $V_g$ for each $g$ !! Thank you for explaining this correspondence!

Answer 1

A Grassmannian $G$ parametrizes the $n$-dimensional subspaces of $W$, and $g\in G$ is the parameter. So you can consider $V_g=V(g)$ to be a function from $G$ to the set of $n$-dimensional subspaces of $W$.

An alternative definition would have $G$ actually be the space of $n$-dimensional subspaces of $W$, and then the subspaces $V_g$ would be the elements of $G$. This can sometimes be a more intuitive definition, but is possibly less helpful if you are trying to build a bundle using $G$.

Answer 2

It's slightly awkward wording: One way to think of this is that the points $g$ in a Grassmannian are the subspaces of $W$ of a given dimension; in this interpretation, $g$ and $V_g$ are just different names for the same object, and we might choose to use the former when we want to emphasize that the object is an element in some set, and the latter when we want to emphasize that the object is itself a vector (sub)space in its own right.