Lemma 9.2.12. The projection map \(\pi \colon V_k(\C ^{\infty }) \to \Gr _k(\C ^{\infty })\) is a principal \(U(k)\)-bundle. Here \(U(k)\) acts on \(V_k(\C ^{\infty })\) by precomposition: given \(A \in U(k)\) and \(\phi \colon \C ^k \hookrightarrow \C ^{\infty }\), we define \(\phi A\) as the composite \(\C ^k \xrightarrow {A} \C ^k \xrightarrow {\phi } \C ^{\infty }\).
Proof. Let \(W \in \Gr _k(\C ^{\infty })\), and define \(U \subseteq \Gr _k(\C ^{\infty })\) to be the subspace of those \(k\)-planes \(V\) for which the orthogonal projection \(W \to V\) is an isomorphism. This is open: its intersection with every finite-dimensional stage \(\Gr _k(\C ^n)\) is open, and \(\Gr _k(\C ^{\infty })\) has the colimit topology. We claim that there exists a \(U(k)\)-equivariant homeomorphism \[ \phi \colon U \times U(k) \xrightarrow {\cong } \pi ^{-1}(U) \] over \(U\), which will suffice to finish the proof. To construct \(\phi \), fix a \(k\)-frame \((w_1, \dots , w_k)\) for \(W\). Given any \(k\)-plane \(V \in U\), the projection \(W \to V\) is an isomorphism, hence sends the \(k\)-frame \((w_1, \dots , w_k)\) to some basis for \(V\). Using Gram-Schmidt orthonormalization (which is a continuous operation), we may turn this into a \(k\)-frame \((v_1, \dots , v_k)\) for \(V\). All in all, we may then define \[ \phi (V,A) := (v_1, \dots , v_k) \circ A. \] Bijectivity of \(\phi \) holds because two orthonormal frames of a \(k\)-plane differ by a unique unitary \((k \times k)\)-matrix. We leave it to the reader to check that this is indeed a well-defined homeomorphism over \(U\). □
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