Math Help - Manifold theory question

1. Manifold theory question

Hello, I've been reading Bishop & Crittenden's Geometry of Manifolds and I'm not sure why they say that F(M,m) is not a linear space where F(M,m) is the set of $C^\infty$ real-valued functions with domain a neighborhood of m (M is a differentiable manifold). Is it because addition of functions fail to be an abelian group due to the imposibility of defining a neutral element?

2. Re: Manifold theory question

Originally Posted by facenian
Hello, I've been reading Bishop & Crittenden's Geometry of Manifolds and I'm not sure why they say that F(M,m) is not a linear space where F(M,m) is the set of $C^\infty$ real-valued functions with domain a neighborhood of m (M is a differentiable manifold). Is it because addition of functions fail to be an abelian group due to the imposibility of defining a neutral element?
Presumably m is supposed to be some fixed element of M?

I think that you have more or less answered your own question. The point seems to be that an element of F(M,m) is not just a function but a pair consisting of a function and its domain. There is no problem defining addition: if f is defined on the neighbourhood U of m, and g is defined on V, then their sum is f+g defined on $U\cap V.$ Similarly, you would want the negative of f to be –f defined on the same domain as f. But what is the neutral element going to be? It should obviously be the zero function, but there is no way to define its domain in such a way that f + (–f) = 0.

For that reason, the usual way to deal with the set of functions that are smooth on some neighbourhood of m is to define an equivalence relation on them, saying that two such functions are equivalent if they agree on some neighbourhood of m. The set of equivalence classes then forms a linear space, the space of germs of smooth functions at m.