Talk:Normed vector space

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Mathematics rating:	Start Class	Mid Priority	Field: Analysis
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. The lead should be an overview, not motivation. The latter should be moved into a separate section. Geometry guy 20:56, 17 June 2007 (UTC)

[edit] Merging

The articles Normed vector space and Linear Algebra/Normed Vector Space need to be merged. (I wasn't aware the latter existed when I wrote the former.) The final article should be placed at Normed vector space, in accordance with the usual naming conventions (and also because it's impossible to add a /Talk to Linear Algebra/Normed Vector Space - the name is too long, apparently).
Zundark, 2001-08-13

As no-one else seemed interested, I did it myself. The other subpages of Linear Algebra should also be moved in due course.
Zundark, 2001-08-16

[edit] Complete ??

Can someone give one more word on the reason why normed vector space is not complete? Since complete normed vector space is Banach space. Thanks. Jackzhp 23:58, 28 October 2006 (UTC)

[edit] Diagram

It is requested that a diagram or diagrams be included in this article to improve its quality. For more information, refer to discussion on this page and/or the listing at Wikipedia:Requested images.

I'm thinking of a diagram with a bunch of vectors radiating from the origin with some depiction of a ruler measuring lengths of vectors and distances from tip to tip measuring. —Ben FrantzDale 21:04, 6 May 2007 (UTC)

I am not sure that this is a great idea. Most work on Normed vector spaces is in infinite dimensional vector spaces. Andrew Kepert 09:44, 7 May 2007 (UTC)

(replying to self...)

Actually, I think the thing that would improve the page is an explicit "Examples" section as appears on similar pages, in particular including some non-complete and non-separated examples. I propose this list should be after the "Definition" section, and be a fairly short list along the lines:

• Any example of a Banach space
• The space c₀₀ of all (Real or Complex) sequences of finite support with supremum norm ||a||_∞ = max_i | a(i) |. This space is not complete, since there are Cauchy sequences which do not converge. For instance the sequence a_n where a_n(i) = 1/i ; if i ≤ n and a_n(i) = 0 otherwise.
• The finite rank operators on a Hilbert space, with operator norm
• The L^p spaces as described below

Some of these could actually be illustrated!

The only other thought I had on a diagram to illustrate this is that feeling for how normed spaces work is via the geometric characterisations of Hahn-Banach, etc, with unit balls, hyperplanes, etc.

Andrew Kepert 10:12, 7 May 2007 (UTC)

[edit] Stupid question?

Cite:

All norms on a finite-dimensional vector space are equivalent from a topological point as they induce the same topology (although the resulting metric spaces need not be the same)

Does it mean that the metric spaces are not homeomorphic?! Or simply they are not isomorphic? The latter, I suppose. Urzyfka —Preceding signed but undated comment was added at 15:44, 23 September 2007 (UTC)

[edit] Seminorm /semi norm consistency

Both forms, "seminorm" and "semi norm" are used in the article. And "semi-normed" also appears. For consistency a single form should be chosen.

The norm article uses "seminorm", so I suggest that.

--84.9.73.5 (talk) 12:24, 1 January 2008 (UTC)