Talk:Quotient space (linear algebra)

what's the significance of Quotient Space?

They're used everywhere. If you want a concrete example, you can use quotient space to define tensor products (let V and W be two finite dimensional vector spaces, let X be (infinite dimensional) free vector space on pairs (v,w), and let X_0 be the subspace spanned by (v,w) + (v,z) - (v,w+z) and (v,sw) - s(v,w) for any v,w,z and scalars s (and likewise for the left component of the pair). The quotient X/X_0 is the (finite dimensional!) tensor product. You get the wedge product if you include (v,w) + (w,v) as well (of course in this case we need V = W). 66.117.135.137 (talk) 09:44, 28 April 2008 (UTC)

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Noting an error in the article

Quote : "Let C[0,1] denote the Banach space of continuous real-valued functions on the interval [0,1]". This is an error, the continuous real-valued functions on the interval [0,1] do form a normed vector space, but do not form a Banach space (since the space is not closed). However I am afraid to fix it myself since I don't know enough on the subject. zermalo (talk) 16:31, 27 April 2009 (UTC)

It depends on the norm. With the L^1 norm it is not complete, but with the supremum norm it is. —Preceding unsigned comment added by MikeRumex (talk • contribs) 09:57, 29 October 2009 (UTC)

Norm on the quotient space

I think that it worth to explicitly note that the infimum in

$\| [x] \|_{X/M} = \inf_{m \in M} \|x-m\|_X.$

is not necessarily attained. However I can't find such an example. Can someone add such an example to the article? bungalo (talk) 15:06, 13 November 2010 (UTC)