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1954•Proceedings of the American Mathematical SocietyRequires access

Algebras of differentiable functions

S. B. Myers

Open publisher page 26 citations

Abstract

1. Let M be a compact differentiable manifold of class Cr, 1 <r < 00, and let Cr(M) be the space of all real functions of class Cr on M. In addition to the obvious algebraic structure of Cr(M), we shall use a normed algebra structure, the norm being obtained by introducing a Riemannian metric on M. The main results (Theorems 1 and 3) will be stated and proved in this section, using lemmas on differentiable manifolds which will be proved in the following s.ection. Theorem 1 is straightforward, Theorem 3 is more difficult.

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What this paper is about

1. Let M be a compact differentiable manifold of class Cr, 1 <r < 00, and let Cr(M) be the space of all real functions of class Cr on M. In addition to the obvious algebraic structure of Cr(M), we shall use a normed algebra structure, the norm being obtained by introducing a Riemannian metric on M. The main results (Theorems 1 and 3) will be stated and proved in this section, using lemmas on differentiable manifolds which will be proved in the following s.ection. Theorem 1 is straightforward, Theorem 3 is more difficult.

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Available abstract

1. Let M be a compact differentiable manifold of class Cr, 1 <r < 00, and let Cr(M) be the space of all real functions of class Cr on M. In addition to the obvious algebraic structure of Cr(M), we shall use a normed algebra structure, the norm being obtained by introducing a Riemannian metric on M. The main results (Theorems 1 and 3) will be stated and proved in this section, using lemmas on differentiable manifolds which will be proved in the following s.ection. Theorem 1 is straightforward, Theorem 3 is more difficult.

Key concepts: Differentiable function, Mathematics, Pure mathematics, Norm (philosophy), Class (philosophy), Section (typography), Manifold (fluid mechanics), Algebra over a field

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