The tangent space to a 𝐶^{𝑘} manifold

Abstract

The algebraic tangent space to a finite dimensional C manifold (1 ^ k ^ oo) is the vector space of linear derivations on C*, the ring of germs of real C functions at p. In this note we give a short proof that, for k < oo, the algebraic tangent space is infinite dimensional. This result is well known but we believe the proof presented here is the easiest. An apparently incorrect proof was given by Papy in [3]. A proof for the case k = 1 was given by Osborn in [2]. A complete solution is given by Newns and Walker in [1], Let ƒ be the maximal ideal of C\. One sees (as in, for example, [4, p. 13]) that the algebraic tangent space is canonically isomorphic to (J//)*. We shall show I/I is infinite dimensional. There is no loss of generality in assuming the manifold is the real line with coordinate x at p. Now if f el, define the order of ƒ by