Let X be a (real or complex) Banach space (not necessarily a Hilbert space), and \(\mathcal {I}(X)\) be the set of all non-trivial idempotents; i.e., bounded linear operators on X whose squares equal themselves. We show that, when equipped with the Banach submanifold structure induced from \(\mathcal {L}(X)\), the subset \(\mathcal {I}(X)\) is a locally trivial analytic affine-Banach bundle over the Grassmann manifold \(\mathscr {G}(X)\), via the map \(\kappa \) that sends \(Q\in \mathcal {I}(X)\) to Q(X), such that the affine-Banach space structure on each fiber is the one induced from \(\mathcal {L}(X)\). Using this, we show that if H is a real or complex Hilbert space, then the assignment

induces a real bi-analytic bijection from the total space of the tangent bundle, \(\mathbf {T}(\mathscr {G}(H))\), of \(\mathscr {G}(H)\) onto \(\mathcal {I}(H)\) (here, \(E^\bot \) is the orthogonal complement of E, \(P_E\in \mathcal {L}(H)\) is the orthogonal projection onto E, and \(T^*\) is the adjoint of T). Notice that this real bi-analytic bijection is an affine map on each tangent plane. Furthermore, if for every \(E\in \mathscr {G}(H)\), we identify \(\mathcal {L}(E,E^\bot )\) with a subspace of \(\mathcal {L}(H)\) via the embedding \(S\mapsto S\circ P_E\), then the inclusion map from \(\mathbf {T}(\mathscr {G}(H))\) to the trivial Banach bundle \(\mathscr {G}(H)\times \mathcal {L}(H)\) is a real analytic immersion. Through this, we give a concrete idempotent in \(M_{n^2}\big (C(\mathscr {G}(\mathbb {K}^n))\big )\) that represents the K-theory class of the tangent bundle \(\mathbf {T}(\mathscr {G}(\mathbb {K}^n))\), when \(\mathbb {K}\) is either the real field or the complex field.

令X为（实数或复数）Banach空间（不一定是希尔伯特空间），而\（\ mathcal {I}（X）\）为所有非平凡等幂的集合；也就是说，X上的平方等于自己的有界线性算子。我们表明，当配备有从诱导的Banach子流形结构\（\ mathcal {L}（X）\） ，子集\（\ mathcal {I}（X）\）是一个局部琐碎解析仿射的Banach捆绑过格拉斯曼歧管\（\ mathscr {G}（X）\） ，经由地图\（\卡帕\）发送\（Q \在\ mathcal {I}（X）\）到Q（X），这样每根光纤上的仿射-巴纳赫空间结构就是由\（\ mathcal {L}（X）\）引起的。使用该函数，我们表明如果H是实或复杂的希尔伯特空间，则赋值

诱导从切丛，总空间中的真实双解析双射\（\ mathbf横置（\ mathscr {G}（H））\） ，的\（\ mathscr {G}（H）\）到\（\ mathcal {I}（H）\）（这里，\（E ^ \ bot \）是E的正交补码，\（P_E \ in \ mathcal {L}（H）\）是到E和\（T ^ * \）是T的伴随。请注意，此实际双解析双射是每个切平面上的仿射图。此外，如果对于每个\（E \ in \ mathscr {G}（H）\），我们将\（\ mathcal {L}（E，E ^ \ bot）\）标识为\（\ mathcal {L }（H）\）通过嵌入\（S \ mapsto S \ circ P_E \），然后从\（\ mathbf {T}（\ mathscr {G}（H））\）到平凡的Banach束\（\ mathscr {G }（H）\ times \ mathcal {L}（H）\）是一个真正的解析沉浸。通过这一点，我们得到在混凝土等幂\（{M_ N ^ 2} \大（C（\ mathscr {G}（\ mathbb {K} ^ N））\大）\）表示该ķ -理论类的当\（\ mathbb {K} \）是实数场或复数场时，切线束\（\ mathbf {T}（\ mathscr {G}（\ mathbb {K} ^ n））\）。