Dan_lior 2014-01-29 23:28:40
A normed space is a vector space with a norm defined on it. That
vector space must be a real or complex vector space, right?
Lrudolph 2014-01-29 23:29:11
firstname.lastname@example.org (Dan Lior) writes:
Well, that would depend on what you mean by “a vector space with
a norm defined on it”, right? That is a matter of convention,
and the merits of competing conventions are something about which
honest and informed people may disagree.
Among the various algebra texts I just pulled off my shelves,
Birkhoff & MacLane, MacLane & Birkoff, and Herstein (at least,
in the editions I possess) all limit their discussions to the
real and complex cases. But Lang doesn’t: he allows the base
field K to be any field with a non-trivial absolute value,
for example, Q or any other subfield of the real or complex
numbers. He doesn’t actualy state any *theorems* unless
K is actually (like R and C, and unlike Q) complete with
respect to the given absolute value. But he certainly is
interested in other complete base fields, like the p-adic
numbers, and normed vectorspaces over those fields. You
don’t have to be, of course (although, _mirabile dictu_,
someone has gotten money to host a conference on “p-adic
physics” this year, so maybe you *should* be).
Robin chapman 2014-01-31 18:37:43
Not necessarily. One can consider normed spaces over p-adic fields
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
“His mind has been corrupted by colours, sounds and shapes.”
The League of Gentlemen
A n niel 2014-02-04 03:22:01
How about a Banach space over the quaternions…
David c. ullri 2014-02-04 03:23:22
That’s usually what’s meant, at least in analysis.
David C. Ullrich
A n niel 2014-02-06 11:27:19
This exercise is for a normed linear space over the real numbers.
If the book fails to say this (perhaps in its definition of normed
space, or of linear space), then you have grounds for complaint.
For example, the inner product in a COMPLEX linear space has some
non-real values, but the one defined above has only real values.
There is a corresponding formula for complex vector spaces, similar to
this one, but with 4 terms instead of 2. After you prove the real
version, you will probably see how to do the complex one.
G. a. edgar 2014-02-11 02:14:22
I have never seen a definition of “inner product” except in the real
and complex cases.
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
Lrudolph 2014-02-11 02:14:31
”G. A. Edgar”
The paper with Math. Reviews number 2000i:46076, to wit,
Ochsenius, H.(RCH-UCCM); Schikhof, W. H.(NL-NIJM),
Banach spaces over fields with an infinite rank valuation,
appears to give such a definition for certain p-adic cases.