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Inner product space

Functional Analysis
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Inner product space


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?
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Tolaso J Kos
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Re: Inner product space


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.
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