What is the maximum value of the inner product of a unit vector and another vector in the same subspace?

Question

My guess, was it would be the infinity norm, but I feel like I am completely wrong.

Answer 1

$$\max_{u : \|u\|=1} \langle u,v\rangle = \|v\|.$$ Cauchy-Schwarz proves the maximum is $\le \|v\|$, and taking $u=v/\|v\|$ shows that the maximum is precisely $\|v\|$.

Answer 2

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.