· Portal  Help Search Members Calendar 
Welcome Guest ( Log In  Register )  Resend Validation Email 
Join the millions that use us for their forum communities. Create your own forum today.  Welcome to Did You Know? 
algebraic topology 
Posted: 15:12 Friday 30 March 2007

Renaissance Group: Friends Posts: 143 Member No.: 14 Joined: 20 Feb 2007 
Did you know? A complete metric space is a metric space in which every Cauchy sequence converges. 
Th.V 
Posted: 07:52 Saturday 14 April 2007


Bronze Age Group: Friends Posts: 21 Member No.: 22 Joined: 19 Mar 2007 
Complete metric space: A metric space, in which every Cauchy sequence is convergent, is a complete metric space. Did you know? Examples include the real numbers with the usual metric, the complex numbers, finitedimensional real and complex vector spaces, (and also the space of squareintegrable functions on the unit interval L˛([0,1]), and the padic numbers). Let us notice however that Q is not a complete metric space, this leads thus to study an extension of Q to a completed space of Q. 

algebraic topology 
Posted: 23:04 Saturday 14 April 2007

Renaissance Group: Friends Posts: 143 Member No.: 14 Joined: 20 Feb 2007 
Did you know? The completeness of the real numbers is often formulated in terms of the leastupperbound property: every set of real numbers that is bounded above has a real least upper bound. It turns out that this is equivalent to the condition that every real Cauchy sequence is convergent. 
JaneFairfax 
Posted: 15:29 Wednesday 08 December 2010

The Enlightenment Group: Moderators Posts: 670 Member No.: 20 Joined: 03 Mar 2007 
Let X be a metric space with metric d. Given a function Did you know? Banach’s fixedpoint theorem states that if X is a complete metric space, any contraction mapping f on X has a unique fixed point, i.e. there exists exactly one point p ∊ X such that 
shygeorge 
Posted: 17:28 Thursday 09 December 2010

The Enlightenment Group: Moderators Posts: 532 Member No.: 161 Joined: 11 Dec 2009 
Did you know? Stefan Banach (1892–1945) was a Polish mathematician.

JaneFairfax 
Posted: 19:33 Thursday 09 December 2010

The Enlightenment Group: Moderators Posts: 670 Member No.: 20 Joined: 03 Mar 2007 
Did you know? Baire’s theorem for metric spaces states that if X is a complete metric space and is a countable collection of nowhere dense subsets of X, is everywhere dense in X. (A subset of a metric space X is nowhere dense iff the interior of its closure is empty; it is everywhere dense iff its closure is the whole space X.) Did you also know? René Baire (1874–1932) was a French mathematician. 
algebraic topology 
Posted: 14:18 Saturday 11 December 2010

Renaissance Group: Friends Posts: 143 Member No.: 14 Joined: 20 Feb 2007 
Did you know? Any metric space can be “completed”. In a metric space X with metric d, let us say that two Cauchy sequences and are equivalent iff 