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Let $V$ be a finite-dimensional inner product space. For $0 \leq d \leq \text{dim}(V)$, define the Grassmannian $G(V, d)$ to be the set of all $d$-dimensional linear subspaces of $V$, equipped with the metric $d(X, Y) = \Vert P_X - P_Y \Vert$, where $P_X, P_Y$ is the orthogonal projector onto $X, Y$, respectively.

I want to use the following fact (to prove other things): The Grassmannian $G(V, d)$ is compact with respect to the metric topology.

I am a physicist. I am quite interested in mathematical questions, but I am not good enough at mathematics to see immediately that the bold statement above is true, so I tried to look it up in the literature. However, I only found this statement either just stated but unproved or the compactness was inferred from from some kind of structures or techniques which I don't know. (I've seen several definitions of the Grassmannian, and I think they are equivalent, but I cannot see this directly.)

Can someone tell me where I can look up a proof of the bold statement above which is understandable with basic knowledge in topological and metric spaces? If the proof is easy enough, I am also happy with a presentation of the proof (idea) here, instead of a reference.

Alternatively, I would also be happy if one could show me a proof for the compactness of $G(V,d)$ in the case where $V$ is a finite-dimensional normed space and \begin{equation} d(X, Y) = \sup\limits_{x \in X, \Vert x \Vert = 1} \inf \{ \Vert x - y \Vert: y \in Y\} \,. \end{equation}

3 Answers 3