  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.
