My guess, was it would be the infinity norm, but I feel like I am completely wrong.
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$$\max_{u : \|u\|=1} \langle u,v\rangle = \|v\|.$$ Cauchy-Schwarz proves the maximum is $\le \|v\|$, and taking $u=v/\|v\|$ shows that the maximum is precisely $\|v\|$.
angryavian
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What is this result called in a broader context (I am thinking Hilbert spaces)? Where can I find more info on it? – plebmatician Dec 19 '17 at 16:18
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@plebmatician Not quite sure you are asking for, but dual norms and Hölder's inequality may be relevant for you – angryavian Dec 19 '17 at 20:14
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In a space $\mathbb R^n$, the inner product between $v$ and $u$ is given by $|u| \cdot |v| \cos \theta$ where $\theta$ is the angle between vectors $v$ and $u$.
The maximum value is attained when $\theta = 0$-- when $u$ and $v$ point in the same direction (and are consequently in the same subspace, no matter which subspace you've chosen). The value is equal to $|u| \cdot |v|$ which is equal to the magnitude of the non-unit vector.
Myridium
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