A multiplier rule in abstract spaces
Herman H. Goldstine · Bulletin of the American Mathematical Society · 1938
The Lagrange multiplier rule has been generalized to certain noncalculus of variations problems by Graves,f Hahn,J and the author.§ Moreover a very general problem was formulated by L. A. Lusternik.|| However his work seems to rest upon a theorem which is stated without proof and which the author is unable to verify. There are also certain other difficulties with his proof. The problem herein presented is so formulated that the problem of Bolza in the calculus of variations, the problems treated by Graves, Hahn, and the author, and the problem of minimizing a functional defined upon an arbitrary Banach space subject to very general numerically-valued side conditions, and numerous other examples are included as special cases. The proof proceeds along lines which are essentially generalizations of the methods of the calculus of variations. This demonstration is made possible by the very powerful implicit function theorems of Hildebrandt and Graves^ and yields analogs of the transversality condition and of the Euler-Lagrange equations. Some instances which explain the number of linear spaces involved are given in the concluding section.