Abstract

Let $H$ be a separable Hilbert space. We consider the Hilbert Grassmannian ${\mathcal{G}}_{\infty }\left(H\right)$ consisting of closed subspaces having infinite dimension and codimension and show that every bijective transformation of ${\mathcal{G}}_{\infty }\left(H\right)$ preserving the class of base subsets is induced by an element of $GL\left(H\right)$ or it is the composition of the transformation induced by an element of $GL\left(H\right)$ and the bijection sending a subspace to its orthogonal complement.

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Mathematical Subject Classification 2000

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