Inner Product in Cauchy's Inequality

Question

I was reading about Cauchy's Inequality and I saw this :

$|u \cdot v|= \|u\| \cdot \|v \| \cdot |\cosθ|$

And I'm confused about usage of $\|u\|$ versus $|u|$. We define inner product as : $u \cdot v = |u| \cdot |v| \cos \theta$ What's the difference between this two conception?

Answer 1

Notice $u$ is a vector in a vector space, say $\mathbb{R}^n$. Thus, the notation $||u||$ means its norm, usually taken to be $\sqrt{ \sum_{k=1}^n u_k^2} $ where $u = (u_1,...,u_2)$. If we take $u,v \in \mathbb{R}^n$, then the function $u \cdot v$ gives an $\mathbf{scalar}$ as an output, thus $|u \cdot v |$ makes sense since $| \cdot |$ is just the absolute value for the field of scalars.

Answer 2

||u|| Is the norm of u (the inner product of u with itself) whereas |u| is actually the absolute value of u. Be careful though because ||u|| and |u| are sometimes used interchangeably to mean norm. You'll see if you prove the inequality it will use the absolute value of u but it looks like here you're just confusing the notation of ||u|| and |u|, they mean the same thing, the norm, so what you saw and how you define inner product are the same thing.

Answer 3

Let's assume that we are working with a real vector space $V$, e.g. $R^3$

Then the inner product $u.v$ of two vectors $u, v \in V$ is a real number, while the "norm" of a single vector is essentially its length (also a real number).

$|x| $ is the notation for the modulus of a real number $x$, i.e. if $x$ is non-negative then $|x| = x$ while if $x$ is negative then $|x| = -x$.

$||v||$ is the notation for then length ("norm") of a vector $v$ and it is always non-negative.

So, stating your formula for the inner product in this notation, $ u.v = ||u||.||v|| cos(\theta)$ which is a real number. The inner product can be negative for $\pi/2 \lt \theta \lt 3\pi/2$, but if we take the modulus, then $|u.v| = |(||u||.||v||cos(\theta)| = ||u||.||v||.|cos(\theta)|$, since the lengths are always non-negative.

(as noted in another answer, the notation can sometimes be confusingly used as $|v|$ for the length ("norm") of a vector, in which case you would write $|u.v| = |v|.|v|.|cos(\theta)|$)