What is the reason norm properties are defined the way they are?

Suppose we have a complex vector space $V$.

A norm is a function $f : V \rightarrow \mathbb{R}$ which satisfies

(i) $f(x) \ge 0$ for all $x \in V$ (Positivity - Non-Negativity)

(ii) $f(x + y) \le f(x) + f(y)$ for all $x, y \in V$ (Subadditivity - Triangle Inequality)

(iii) $f(\lambda x) = |\lambda|f(x)$ for all $\lambda \in \mathbb{C}$ and $x \in V$ (Positive Homogeneity)

(iv) $f(x) = 0$ if and only if $x = 0$ (Definiteness)

From these properties, we define the $l_p-norm$

$$||x||_p = \left(\sum \limits_{i=1}^{n}|x_i|^p \right)^\frac{1}{p}$$

My questions relate to why these four properties were chosen for this definition:

(1) If norm is just a measure, why do we need all four of these properties?

(2) Do these properties allow for a nice geometric interpretation of norms?

(3) Were these properties chosen to nicely allow for a nesting of the various norms? For example

$$||x||_\infty \le ||x||_2 \le ||x||_1$$

(4) Were these properties chosen to satisfy pairwise inequalities? For example $$||x||_1 \le n||x||_\infty$$

(5) Were these properties chosen because they lend themselves in other variants? For example, in $l_2$ for bi-infinite sequences, vector-valued sequences or general sequences.

(6) Do these properties allow an extension into other function spaces where all of the items above still hold? For example, $L_2$ or $L_p$ function spaces.

I haven't looked at norms in quite a while, so please forgive me if I am missing something very obvious.

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I'll have to find my link to a math etymology site, which addresses first uses and subsequent uses of terms, when they were introduced, by whom, to represent what, etc. ... If I remember, I'll post a link here to it. But yes, I always wonder about how/when/why "conventions", terms, notation, ideas, first came into play, and how they've morphed over time. – amWhy Apr 19 '13 at 2:33
@amWhy: Those are some excellent questions and it would be nice to have something that captures all of that, as well as a detailed list of notation and usage over time. – Amzoti Apr 19 '13 at 2:34

The intuition behind the definition of norm is that the norm of a vector is a way to measure the length of the vector. Then the definition becomes clear:

• The length should be non-negative.
• In a triangle, the length of one side is smaller than the sum of the lengths of the other sides (this where the name triangle inequality comes from).
• If we stretch a vector, i.e. if we multiply it by a number $\lambda$, the length should be multiplied by $|\lambda|$.
• The only vector with length $0$ should be the zero vector.

Moreover, in a normed space you can define a distance as $$d(x,y)=\|x-y\|.$$ This gives the space a structure of metric (and hence topological) space, allowing the definition of open and closed sets, limits,...

There are many normed spaces used throughout mathematics:

• $\mathbb{R}^n$ and $\mathbb{C}^n$.
• The space of continuous functions $f\colon K\to\mathbb{C}$ where $K$ is a compact topological space with the uniform norm.
• The $L^p$ and $\ell^p$ spaces, $1\le p\le\infty$.
• Sobolev spaces.
• ...
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A possible motivation for three of the four properties begins with the idea that $\Vert x\Vert$ should be the distance between the points $0$ and $x$, or equivalently, the length of the vector $x$. Now the vector from the point $x$ to the point $y$ is $y-x$, so $\Vert y-x\Vert$ should be the distance between those two points. That means the function $x,y\mapsto\Vert y-x\Vert$ should be a metric on the vector space. In order for it to satisfy the axioms of a metric, you need properties 1, 2, and 4 on your list.

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