3

While we define norm on a vector space, we consider only real or complex vector field. But can we generalize this norm on a vector space over an arbitrary field ? I think this can be done, but we have to define a suitable modulus function on the ground field to be meaningful in the property ||cx||=|c|||x||, what |c| means. Is this possible to define norm in this general situation? And why we only bother about real or complex vector space?

SJA
  • 174
  • For sure! In number theory, we often consider vector spaces over $\mathbb{Q}$, and we can define $|c|$ as the absolute value for all scalars $c \in \mathbb{Q}$. I'm not sure whether this can be extended to finite fields though ... I'll leave that for others to answer. – Adithya Chakravarthy Aug 31 '20 at 18:42
  • 1
    From wikipedia: "Other choices are possible. For example, for a vector space over $\mathbb {Q}$ one could take $|\alpha |$ to be the $p$-adic norm, which gives rise to a different class of normed vector spaces. – Dietrich Burde Aug 31 '20 at 18:43

2 Answers2

0

Let $F$ be a field and $V$ a vector space over $F$. Define $|\cdot |: F \to \mathbb{R}$ by $|f| = 1$ for $f \ne 0 $ and $|0| = 0$. Define $\| \cdot \| : V \to \mathbb{R}$ by $\|v \| = 1$ for $v \ne 0$ and $\| 0\| = 0$.

Then $\|v\| \geq 0$, $\|v\| = 0\iff v =0 $, if $\lambda \ne 0$ and $v \ne 0$, then $\lambda v \ne 0 $ and $\|\lambda v \| = 1 = |\lambda| \|v\|$, otherwise $\lambda v = 0$ and $|\lambda| \|v\| = 0$, so $\|\lambda v\| = |\lambda |\|v\|$ either way.

Finally as long at least one of $v$ or $w$ is nonzero, then $\|v + w\| \leq 1 \leq \|v\| + \|w\|$, else $v = w = 0$, so $\|v+w\| = \|0\| =0 \leq 0 + 0 = \|v\| + \|w\|$.

So we have define a not very interesting norm on an arbitrary vector space over an arbitrary field. Note we need to define a norm on the field, which should follow the positive definiteness requirement as well as triangle inequality (which ours does) and maybe also a multiplicativity requirement: $|ab| = |a||b|$ (which ours does also).

Finally, as was commented, there are more interesting norm on number fields in number theory. But this shows it always possible in some sense.

Physical Mathematics
  • 4,453
0

We can (at least sometimes) even define a norm on a module over a ring, a notion which generalizes that of a vector space.

One interesting example is the Gaußian integers, $\Bbb Z[i]$. An arbitrary element looks like $a+bi, a,b\in\Bbb Z$.

If you define the "norm" by $N(a+bi)=a^2+b^2$, it has the nice property of being multiplicative.