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!