Universal vector bundle over the infinite Grassmannian

Definition 9.2.2. For \(n\geq k\), let \[ E_{k,n}:=\{(V,x)\in \Gr _k(\C ^n)\times \C ^n\mid x\in V\} \] be the canonical rank \(k\) bundle over \(\Gr _k(\C ^n)\). We define \[ E_{\univ }:=\bigcup _{n\geq k}E_{k,n} \] with the colimit topology, and let \(p_{\univ }\colon E_{\univ }\to \Gr _k(\C ^\infty )\) be the projection \((V,x)\mapsto V\).