(b) Let A and B be two subset of a linear space X such that ACB, show thatwhether if A is affine set then B affine or need not and if B affine set then A affine setor need not.Qz/antonly bea-Show that every hyperspace of a vecor space X is hyperplane but the conversneed not to be true.b- Let M be a finite dimension subspace of a Banach space X show that M is closed set.c-Show that every two norms on finite dimension vector space are equivant (1)Q/answer only twoa-Write the definition of bounded set in: a normed space and write with prove anequivalent statement to a definition.b- Let f be a function from a normed space X into a normed space Y, show that fcontinuous iff f is bounded.c-Show that every finite dimension normed space is a Banach.Q/a- Let A and B two open sets in a normed space X, show that by definitionAnB and AUB are open sets.(1noodtrue

Answered: Image /qna-images/answer/3cd7e20e-8612-4235-a58d-22bc2c3acf31.jpg

Answered: (b) Let A and B be two subset of a…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.2: Vector Spaces

Problem 38E: Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1)...

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Similar questions

Let V be the set of all positive real numbers. Determine whether V is a vector space with the operations shown below. x+y=xyAddition cx=xcScalar multiplication If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.
Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}
Give an example showing that the union of two subspaces of a vector space V is not necessarily a subspace of V.
Let v1, v2, and v3 be three linearly independent vectors in a vector space V. Is the set {v12v2,2v23v3,3v3v1} linearly dependent or linearly independent? Explain.
Let V be an two dimensional subspace of R4 spanned by (0,1,0,1) and (0,2,0,0). Write the vector u=(1,1,1,1) in the form u=v+w, where v is in V and w is orthogonal to every vector in V.
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.
Show that the set of all pairs of real numbers (x, y) with the operations (X1, Y1) + (x2, Y2) = (x1 + x2 + 1, y1 + Y2+ 1) and k (*1, Y1) = (kx1, kyı) is not a vector space.
8. (. you list are all different from each other and from P7. ) List four vector spaces that are isomorphic to P7. Please make sure that the vector spaces
Determine whether the set R? with the operations (x1, y1) + (x2, y2) = (x,x2, Y¡Y2) and c(x1, y1) = (cx1, cyı) is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.
Give an example of 3 vector spaces that are not Rn. Explicitly state thedefinition of additon and the zero vector in each space. There is a solution on this site but it is hard to read and understand.
If (x1,y1) + (x2,y2) = (x1x2,y1y2) and c(x1,y1)= (cx1,cy1) are vector spaces, verify if both are a vector space property and identify the properties that it will fail. (Use the 8 theorems)
an Given that { 91, 92, 93, 94} is a linearly independent set of vectors in av Vector space space V'( note that V need not be IR"). Use the diff Prove that {a an ax; -91 +92, 91-92 +93 ;-9, +92-93 +94"} is also a independence) linearly independent set (use the definition of Linearly

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