Re: ? normed space and metric space
 From: David C. Ullrich <dullrich@xxxxxxxxxxx>
 Date: Tue, 19 Feb 2008 04:51:24 0600
On Mon, 18 Feb 2008 08:28:44 0500, "Cheng Cosine"
<acosine@xxxxxxxxxx> wrote:
"David C. Ullrich" <dullrich@xxxxxxxxxxx> wrote in message
news:thtir39ngom5fo2k9vth1pc3avuolerdgr@xxxxxxxxxx
On Mon, 18 Feb 2008 02:43:52 0500, "Cheng Cosine"
<acosine@xxxxxxxxxx> wrote:
...
What are the differences between a normed space and a metric space?
I thought a metric space means a space that comes with the "distance"
being defined, and likewise a normed space with a norm being defined.
But isn't norm kind of distance? Any differences?
Any normed space is indeed a metrix space, but not conversely.
First, we only talk about norms on _vector spaces_, and not every
metric space is a vector space. But even for vector spaces there's
a big difference  there exist metrizable vector spaces which are not
normable.
Thanks you two so that I understand a little bit more :)
So from "large" to "small" (larger contains smaller):
normed space > metric space > inner product space
Correct?
No.
But in reading defs of the above spaces, I am confused with a complete
space
and a compact space. What are their differences?
??? Read the definitions more carefully  they're simply
not the same thing at all.
...
For example, let C be the space of continuous functions on R.
For f, g in C and n a positive integer define
d_n(f,g) = sup{f(t)  g(t) : t <= n}.
Define
d(f,g) = sum (2^{n} d_n(f,g)/(1+d_n(f,g)).
Then d is a metric on C, but there is no norm on C which gives
the same topology.
Well, I still don't see why defining a metric d by the above leads to
nonexistence of norm.
Also, per Elliot's post:
A metric space (S,d) is a set S and a function d:S^2 > R with
d(x,y) = 0 iff x = y
d(x,y) = d(y,x)
d(x,z) <= d(x,y) + d(y,z)
But the def you used for your d(f,g) involves an integer n, doesn't
that mean that you need another set to place n? However, a metric,
according to Elliot's post, only involves 1 set. What's goning on then?
Thank you,
by Cheng Cosine
Feb/18/2k8 NC
David C. Ullrich
.
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 ? normed space and metric space
 From: Cheng Cosine
 Re: ? normed space and metric space
 From: David C . Ullrich
 Re: ? normed space and metric space
 From: Cheng Cosine
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