## Lagrangian Subspaces: Intersection Theory, and the Morse Index Theorem in Finite Dimensions |

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### Contents

Linear Symplectic Structures on Banach | 5 |

9 | 39 |

The Fredholm Lagrangian Grassmannian | 50 |

Copyright | |

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### Common terms and phrases

admits Banach manifolds Banach space boundary closed compact operators complement completes the proof conjugate index conjugate points continuously decreasing family curve of lagrangian Darboux decomposition defined definition denotes dense diffeomorphic differential operator dim Ker eigenvalue eigenvalue curves elliptic operators F x F fibre bundle finite dimensional finite dimensions follows Fredholm lagrangian grassmannian GL(E GL(n Graph H x H hence Hilbert space homeomorphic homotopy equivalence identified integer intersection number intersection theory kernel lagrangian grassmannian lagrangian subspaces Lemma linear symplectic structure Maslov Cycle Maslov Index Morse Index Morse Index Theorem number of conjugate open set orbit Proposition Q.E.D. Corollary Q.E.D. Remarks result Schauder basis Section 1.1 self-adjoint operator sequence simply Sp(E Sp(n splits strictly increasing subgroup submanifold subset Suppose symmetric bilinear form symmetric form symplectic form symplectic group symplectic space symplectic structure tangent space topological Uhlenbeck 58 vector bundle x,y e zero