Fredholm-Lagrangian-Grassmannian and the Maslov index

We explain the topology of the space, so called, Fredholm-Lagrangian-Grassmannain and the quantity Maslov index'' for paths in this space based on the standard theory of Functional Analysis. Our standing point is to define the Maslov index for arbitrary paths in terms of the fundamental spectral property of the Fredholm operators, which was first recognized by J. Phillips and used to define the Spectral flow’'. We tried to make the arguments to be all elementary and we summarize basic facts for this article from Functional Analysis in the Appendix.