# Is a norm unique in its defined space?

Keeping the definition of norms in mind, given where a norm is defined (like the Euclidean space, or on a set of functions (like mappings from $$C^{\infty} \to C^{\infty}$$), or any other valid space) unique?

In other words, for instance, are the norms $$||K|| = \sup_{I \subset R^n}(f - g) ; \ \ f,g:R^n \to R^m;\ \ f,g \in C^{\infty}$$ or $$||K||=\sqrt{x^2 + y^2}, \ \ x,y \in R \$$ unique where they are applied? Is there any other norm that passes the three properties of norms and yet different from ones I've mentioned?

On an infinite dimensional vector space, there exist norms which are not equivalent (in the sense described here). Examples of such norms are the $$p$$-norms. For a function $$f: \mathbb{R} \rightarrow \mathbb{R}$$ s.t. the following integral exists, the $$p$$-norm with $$1 \leq p < \infty$$ is defined by $$(\int \vert f(x) \vert^p dx )^{1/p}$$. For any two different $$p$$ these norms are not equivalent.
• It depends on what you mean by two norms being equal. Usually, one means that two norms are equal, if they induce the same topology. On a finite dimensional (real) vector space, this is the case. I.e. any two norms that are defined on $\mathbb{R}^n$ induce the same norm. For an infinite dimensional vector space (like the space of continuous real valued functions) this is not the case. Oct 24 '19 at 20:53