This is a follow-up question to this.
In my adaptive control book, we are using the following inequality. $$\left\|\Phi^T\left(\mathbf{x}\right)\Phi\left(\mathbf{x}\right)\right\| \geq \left\|\Phi\left(\mathbf{x}\right)\Phi^T\left(\mathbf{x}\right)\right\|$$
where $\Phi\left(\mathbf{x}\right) \in \mathbb{R}^m$ and are "implicit" functions of time due to $\mathbf{x}\left(t\right)$. In the previous question, they are actually equal if the norm is Euclidean ($p = 2$). Since a control norm is a signal norm in essence, how does one prove that the inequality is true?