Dan_lior2014-01-29 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?

dan

Lrudolph2014-01-29 23:29:11

dan_lior@hotmail.com (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).

Lee Rudolph

Robin chapman2014-01-31 18:37:43

Not necessarily. One can consider normed spaces over p-adic fields

for instance.

—

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 niel2014-02-04 03:22:01

How about a Banach space over the quaternions…

David c. ullri2014-02-04 03:23:22

That’s usually what’s meant, at least in analysis.

************************

David C. Ullrich

A n niel2014-02-06 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. edgar2014-02-11 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/

Lrudolph2014-02-11 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.

Lee Rudolph

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