These are the initial notes that were typed for Magnus R. Hestenes by Thelma Harvey in the Mathematics Department at UCLA. They were used for Mathematics 285G, Seminar in Analysis, Winter and Spring Quarters, 1967. What is presented here includes some minor corrections and editorial remarks describing how some of this theory was developed and later changed. Quadratic Form Theory and Differential Equations by John Gregory, Academic Press, 1980, is a comprehensive treatment of this subject with Chapter Two, Abstract Theory, of special interest. Applications of the theory of quadratic forms in Hilbert space in the calculus of variations by Magnus R. Hestenes, Pacific Journal of Mathematics, volume 1, pp. 525-582 {1951), forms a basis for this subject and was a development which extended the ideas of Marston Morse, George D. Birkhoff, and Henri Poincare. The subject matter presented here is algebraic in nature and includes an infinite dimensional local Morse Theory. Some of the indices include signature or negative index, nullity, and relative nullity. The idea of a Q-closed subspace is defined by two conditions at the beginning of section 9 of these notes and later changed to one condition by Magnus R. Hestenes as described in John Gregory's book by equation {12), page 65. Also, the term manifold in both Hestenes' notes and Gregory's book means linear subspace.

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