(c) For the remaining parts of this problem, assume that (E, ||- ||) isa real normed vector space, and that the norm || . || satisfies theparallelogram identity.Prove that in this case, for all x, y, z € E,|||x+y+z||² − ||x+y−z||² = ||x+z||² — ||x−z||²+||y+z||² — ||y−z|| ².(Hint. One possible approach starts by applying the parallelo-gram identity to each term on the left hand side. For example,|||x+y+z||² = 2||x + z||² + 2||y||² − ||(x + 2) − y||².Use this idea to show that||x+y+z||²||x+y-z||² = 2||y||²—2||x||²+2||x+z||²—2||y-z||².Argue that the same identity holds with and y interchanged,by symmetry. and then add the two equations.)

Answered: Image /qna-images/answer/c41eaf9d-dbc2-46b9-98dc-49f90cab6bba.jpg

Answered: (c) For the remaining parts of this…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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(c) For the remaining parts of this problem, assume that (E, ||- ||) is a real normed vector space, and that the norm || - || satisfies the parallelogram identity. Prove that in this case, for all x, y, z = E, ||x+y+z||²—||x+y−z||² = ||x+z||²—||x−z||²+||y+z||² − ||y−z||².