Definition of norm

Question

This is probably a simple one. Defining a normed vector space over a field K we ask the norm function to satisfy the equality: $||\alpha x||=| \alpha | \ ||x||$. However, if K is not a field of reals or complex numbers, it is unclear what $|\alpha|$ stands for.

Does this mean we ALWAYS have to first define a norm (an absolute value) $|\alpha|$ on K to define a norm on the vector space over K?

Thank you!

score 1 · Answer 1 · answered Dec 25 '13 at 15:36

1

Yes. Having a norm in your vector space implies a notion of length; so you need to have a meaning for how the length of a vector is changed when multiplied by a scalar.

answered Dec 25 '13 at 15:36

Martin Argerami

205,756

score 1 · Answer 2 · answered Dec 26 '13 at 16:10

1

In functional analysis, it is uncommon to consider vector spaces over fields that are not subfields of the field $\mathbb{C}$.

answered Dec 26 '13 at 16:10

kharvd

160

Definition of norm

2 Answers2