$C^{0}$-norm of a map

Question

I have the following question: Consider a continous map $f:M\rightarrow N$, where $M,N$ are smooth manifolds. How can one define its $C^{0}(M,N)$-norm, i.e. $||f||_{C^{0}(M,N)}$ ?

Greetings, Daniel

Answer 1

A norm doesn't make sense, in the first place, unless you have a vector space. Unless $N$ is a vector space, you won't have a vector space structure on $C^0(M,N)$. If it is a normed vector space, then you'll get an induced norm by the same construction as below.

However, when $M$ is compact and $N$ has a metric space structure $d_N$ (e.g., induced by an embedding in $\Bbb R^k$ for some $k$ or by a Riemannian metric) you can define a metric on $C^0(M,N)$ by setting $$d(f,g) = \max_{x\in M} d_N(f(x),g(x))\,.$$ In general, there are several natural $C^k$ topologies on the space $C^k(M,N)$ of $C^k$ maps from $M$ to $N$. See, for example, Morris Hirsch's Differential Topology.