devotion & mathematics

For a video explanation, click the link 👉 https://youtu.be/Ai7fvMYv4Jo Theorem :            The intersection of subspaces of  a vector space is again a subspace of the vector                space                                           Or          If    W 1  and  W 2  be any two subspaces of a vector spave V(F) then    W 1 ∩ W 2    is also             a subspace of V(F)  Proof :                                                     * Linear Sum of Subspaces   * What is a vector space  * Theorem on vector space      * Historical Introducti...

  CONTENTS : Theorem  * : Cantor's  Intersection Theorem Theorem ** :  Let {x n } be a sequence in a metric space (X,d). For each positive integer n, write  E  n ={x  m /n ≥ m}. Then { x  n } is a Cauchy sequence in X if and only if lim diamE n =0. Theorem *** :  For any subset E of a metric space (X,d), diam E=diam Ē. The following are the proofs of the above statements : proofs : Theorem * : Cantor Intersection Theorem  Statement & Proof :     Theorem **  :  Let {x n } be a sequence in a metric space (X,d). For each positive integer n, write  E  n ={x  m /n ≥ m}. Then { x  n } is a  Cauchy sequence  in X if and only if lim diamE n =0. Proof :   Theorem ***  :  For any subset E of a metric space (X,d), diam E=diam Ē. Proof :                    people also ask :  1. What is a metric space?  2. What i...

 PROOFS OF FEW RESULTS  1:                    Here we are going to see the proofs of some results what we know. They are   1.  Intersection of an infinite collection of open sets is not open.  2. The set N of natural numbers is not open.  3.  Every neighborhood is an open set.  4.  Union of any collection of open sets is open. Now come to proofs : 1. INTERSECTION OF AN INFINITE COLLECTION OF OPEN SETS IS NOT OPEN : PROOF : 2. THE SET N OF NATURAL NUMBERS IS NOT OPEN: PROOF :  3. EVERY NEIGHBORHOOD IS AN OPEN SET : PROOF : 4. UNION OF ANY COLLECTION OF OPEN SETS IS OPEN : PROOF : people also ask         * what is an interior point * what is an open set        * what is a neighborhood                * what is extended real number                      *...

  OPEN & CLOSED SETS In this session we see open & closed sets with their properties. OPEN SET : -      A non empty set A in the real line R is called an open set , every point in A is an interior point of A.                                                                     Examples:- * (a, b)  is an open set.  * [a, b] is not open since the end points a, b are not interior points. * Every finite set is not open. * The sets N, Z and Q are not open sets. * The set R of all real numbers is open. * The set of complex numbers |z|<1 is open. * The set of complex numbers |z|  ≤1 is not open. * Every neighbourhood is an open set. Properties of open sets:- 1. Every open set is a union of open intervals. 2. Union of two open sets is open. 3. Intersection of two open sets is ...