Inner product space

Functional Analysis
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Joined: Sun May 31, 2020 4:26 pm

Inner product space


Post by aristarty »

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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Location: Larisa

Re: Inner product space


Post by Tolaso J Kos »

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.
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