4. If x,y are nonzero orthogonal vector in X, then x,y) is linear independent.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![4. If x,y are nonzero orthogonal vector in X, then {x,y) is linear independent.
5. If xx and y 1x, for all n, then xly
"
6. M = (x = (x) ef:x₂=0, neN} is closed subspace of ² and find M
7. Find M if M = [(e₁,e₂,e)], where e, = (6)
8. Let X R². Find M if
a.. M = {x}
b. M = {x₁,x₂} and M is linear independent.
9. If X is finite dimensional Hilbert space, then it has Orthonormal basis.
10. Let {x} be an Orthonormal sequence in a pre-Hilbert space X and let x € X. Show that
x-y LM, where y=[^,x, and M =[[xxx}].
11. Show that In a finite dimensional normed space, each closed and bounded set is compact.
12. Let A be a subset of a Hilbert space X . Show that
a.. A = A
b.. A¹ = [4]
c.. A is dense in X iff 4+ = {0}
13. Let X,Y are Hilbert space on a field F. Show that X,Y are Hilbert Isomorphic iff
dim X = dim Y](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4725b67c-7cdd-4a39-ac18-c5ec8e35aaaa%2F5414224d-d089-4297-b7a4-9f83bca3d2c1%2Fznbzu45_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. If x,y are nonzero orthogonal vector in X, then {x,y) is linear independent.
5. If xx and y 1x, for all n, then xly
"
6. M = (x = (x) ef:x₂=0, neN} is closed subspace of ² and find M
7. Find M if M = [(e₁,e₂,e)], where e, = (6)
8. Let X R². Find M if
a.. M = {x}
b. M = {x₁,x₂} and M is linear independent.
9. If X is finite dimensional Hilbert space, then it has Orthonormal basis.
10. Let {x} be an Orthonormal sequence in a pre-Hilbert space X and let x € X. Show that
x-y LM, where y=[^,x, and M =[[xxx}].
11. Show that In a finite dimensional normed space, each closed and bounded set is compact.
12. Let A be a subset of a Hilbert space X . Show that
a.. A = A
b.. A¹ = [4]
c.. A is dense in X iff 4+ = {0}
13. Let X,Y are Hilbert space on a field F. Show that X,Y are Hilbert Isomorphic iff
dim X = dim Y
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