Welcome to mathimatikoi.org forum; Enjoy your visit here.

Inner product space

Functional Analysis
Post Reply
aristarty
Articles: 0
Posts: 1
Joined: Sun May 31, 2020 4:26 pm

Inner product space

#1

Post by aristarty » Sun May 31, 2020 4:33 pm

Hi everyone!
I faced with a problem: prove that two vectors of inner product space is on the same ray only when $\left \| x+y \right \| = \left \| x \right \| + \left \| y \right \|$.
Does anyone know how to prove it?
User avatar
Tolaso J Kos
Administration team
Administration team
Articles: 2
Posts: 855
Joined: Sat Nov 07, 2015 6:12 pm
Location: Larisa
Contact:

Re: Inner product space

#2

Post by Tolaso J Kos » Tue Jun 09, 2020 11:33 am

Hint: Equality holds when vectors are parallel i.e, $u=kv$, $k \in \mathbb{R}^+$ because $u \cdot v= \|u \| \cdot \|v\| \cos \theta$ when $\cos \theta=1$, the equality of the Cauchy-Schwarz inequality holds.
Imagination is much more important than knowledge.
Post Reply