t the ordered pairs in the equivalence relations produced by these partitions of { a , b , c , d , e , f , g } . a) { a , b } , { c , d } , { e , f , g } b) { a } , { b } , { c , d } , { e , f } , { g } c) { a , b , c , d } , { e , f , g } d) { a , c , e , g } , { b , d } , { f } A partition P 1 is called a refinement of the partition P 2 if every set in P 1 is a subset of one of the sets in P 2 .
t the ordered pairs in the equivalence relations produced by these partitions of { a , b , c , d , e , f , g } . a) { a , b } , { c , d } , { e , f , g } b) { a } , { b } , { c , d } , { e , f } , { g } c) { a , b , c , d } , { e , f , g } d) { a , c , e , g } , { b , d } , { f } A partition P 1 is called a refinement of the partition P 2 if every set in P 1 is a subset of one of the sets in P 2 .
Solution Summary: The author explains the equivalence relations produced by lefta,b,c,d,e,f,gright.
Q1: A: Let M and N be two subspace of finite dimension linear space X, show that if M = N
then dim M = dim N but the converse need not to be true.
B: Let A and B two balanced subsets of a linear space X, show that whether An B and
AUB are balanced sets or nor.
Q2: Answer only two
A:Let M be a subset of a linear space X, show that M is a hyperplane of X iff there exists
ƒ€ X'/{0} and a € F such that M = (x = x/f&x) = x}.
fe
B:Show that every two norms on finite dimension linear space are equivalent
C: Let f be a linear function from a normed space X in to a normed space Y, show that
continuous at x, E X iff for any sequence (x) in X converge to Xo then the sequence
(f(x)) converge to (f(x)) in Y.
Q3: A:Let M be a closed subspace of a normed space X, constract a linear space X/M as
normed space
B: Let A be a finite dimension subspace of a Banach space X, show that A is closed.
C: Show that every finite dimension normed space is Banach space.
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RELATIONS-DOMAIN, RANGE AND CO-DOMAIN (RELATIONS AND FUNCTIONS CBSE/ ISC MATHS); Author: Neha Agrawal Mathematically Inclined;https://www.youtube.com/watch?v=u4IQh46VoU4;License: Standard YouTube License, CC-BY