Geodesic
On Isomorphisms of Lattices of Closed Subspaces
Canadian journal of mathematics, Tome 36 (1984) no. 5, pp. 820-829

Voir la notice de l'article provenant de la source Cambridge University Press

By a projectivityof vector spaces Xand Yover fields F and G is meant an isomorphism Ψ:(X) → (Y) of their lattices of subspaces. A basic theorem of projective geomtry [2, p. 44] asserts that, for spaces of dimension at least 3, any such projectivity is of the form Ψ(M) = SM for a bijection S:X → Y which is semi-linear in the sense that S is an additive mapping for which there exists an isomorphism σ:F→ G such that S(λx) = σ(λ)Sx for all λ ∈ Fand all x∈ X.In [12] Mackey obtained a continuous version of this result: for real normed linear spaces Xand Y, the lattices and of closed subspaces are isomorphic if and only if X and Yare isomorphic (i.e., via a bicontinuous linear bijection). 
Fillmore, P. A.; Longstaff, W. E. On Isomorphisms of Lattices of Closed Subspaces. Canadian journal of mathematics, Tome 36 (1984) no. 5, pp. 820-829. doi: 10.4153/CJM-1984-048-x
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