Does Cauchy-Schwarz hold for: $ \langle\textbf{u},\textbf{v}\rangle \;\leq ||\textbf{u}|| \cdot ||\textbf{v}|| $

Question

I am wondering whether the Cauchy-Schwarz inequality does hold when absolute value is not considered for the LHS.

Let me explain: In standard Cauchy-Schwarz we have:

$| \langle \textbf{u},\textbf{v}\rangle |\;\leq \|\textbf{u}\| \cdot \|\textbf{v}\| $

But will this hold if no absolute value is taken for the left hand side?

My thoughts:

Since $| \langle\textbf{u},\textbf{v}\rangle |\; = \|\textbf{u}\| \cdot \|\textbf{v}\| \cos(\alpha) $

we have that since $\cos(\alpha) \leq 1$, then the inequality:

$ \langle\textbf{u},\textbf{v}\rangle \;\leq \|\textbf{u}\| \cdot \|\textbf{v}\| $, should hold.

May I get your opinion, thoughts? please!

Answer 1

Note that $\langle u,v\rangle\leqslant\bigl\lvert\langle u,v\rangle\bigr\rvert\leqslant\lVert u\rVert.\lVert v\rVert$.