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For what category of function it’s a norm?

Recently I’m studying a textbook of functional analysis. In that book, it says that the class of the functions which are bounded, continuous and a sub-linear function could be a norm. But I don’t find that true since the definition of a norm requires that it’s a norm in $\mathbb{R}^n$ with $n \geq 1$. However, I’m also not sure if that’s true.


You can define the set of normal numbers as $ \mathbb{R}^1_{+} := \{x \in \mathbb{R}^1 \mid \exists \, \epsilon >0 \textrm{ s.t.} \, x-\epsilon 0 \textrm{ s.t. } x\in \mathbb{R}^1_{+} \}$$
for every $x \in \mathbb{R}^1$.


Yes, a norm can be defined on any vector space, and it’s usual to restrict attention to the case that the vector space is finite-dimensional (so the norm is no longer real-valued, but a function from the vector space into $\mathbb{R}$). In this case, it turns out that the only norm on $\mathbb{R}^n$ which takes finitely-additively-measurable sets into non-negative real numbers is the $L^p$ norm, in which case the only reasonably-behaved functions in that class are the “regular” Lebesgue measurable functions $f$ such that $\int |f|^p$ is finite. If we define a norm using regular Borel sets as in your question, it turns out that “every” function which is continuous and non-negative and measurable is not a norm, at least not in the usual sense of the word. This is a famous open problem.


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