Let V be a normed vector space (real or complex valued) with norm $\|\cdot\|_V$. For any nonempty and bounded subset $A \subseteq V$ one can define $\|A\|$ via $$\|A\|:=\sup\{|x|:x\in A\}$$

I noticed, that $\|\cdot\|$ fulfills similar properties as a norm:

  • $\|A\| = 0 \Leftrightarrow A = \{ 0 \}$
  • $\|\lambda A \| = |\lambda|\cdot \|A\|$
  • $\|A+B\| \le \|A\|+\|B\|$

Thereby I define

  • $\lambda A := \{\lambda x: x \in A \}$
  • $A+B:= \{x+y:x\in A,y\in B\}$

and $\lambda$ is a real or complex number (depending on whether $V$ is real or complex valued).

But $\|\cdot\|$ is no norm, because the underlying set of bounded, nonempty subsets of $V$ is no vector space (for example we have $A-A \neq \{0\}$ if $A$ has more than one element).

Is there a concept for $\|\cdot\|$ in mathematics? Is there a concept for something which behaves like a norm but the underlying set does not need to be a vector space?

  • $\begingroup$ you would want a pointed set and also addition as well as scalar multiplication. So something like a module, maybe without any torsion. $\endgroup$ – Daniel Valenzuela Dec 16 '14 at 15:32

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