PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (211)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (36)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
[parent] yet another proof of parallelogram law (Proof)

Define $ g(\epsilon) = \langle x+\epsilon y , x + \epsilon y\rangle$, where $ \epsilon$ is real. Then $ g(\epsilon) = \langle x,x\rangle + \epsilon (\langle y,x\rangle + \langle x,y\rangle) + \epsilon^2 \langle y,y\rangle .$ Hence,

$\displaystyle \Vert x + y \Vert ^2 + \Vert x - y \Vert ^2 = g(1) + g(-1) = 2\langle x,x\rangle + 2\langle y,y\rangle = 2 \Vert x \Vert ^2 + 2 \Vert y \Vert ^2.$



"yet another proof of parallelogram law" is owned by Mathprof.
(view preamble)

View style:


This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: real

This is version 2 of yet another proof of parallelogram law, born on 2006-08-03, modified 2006-08-04.
Object id is 8214, canonical name is YetAnotherProofOfParallelogramLaw.
Accessed 856 times total.

Classification:
AMS MSC: 46C05 (Functional analysis :: Inner product spaces and their generalizations, Hilbert spaces :: Hilbert and pre-Hilbert spaces: geometry and topology )
Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add example | add (any)