I have encountered the above inequality in a proof, where
- $u(\tau) \in \mathbb{R}^{n\times m}$, and $w \in \mathbb{R}^n$ is any unit vector
- $T_0$ and $t$ is are positive real numbers
The proof states that the inequality follows from the Cauchy-Schwarz inequality; however, it is very different from the following standard Cauchy-Schwarz inequality:
$$\int_{a}^b |f(x)|^2 dx \int_{a}^b |g(x)|^2 dx \geqslant \big|\int_{a}^b f^\star(x)g(x) dx \big|^2.$$
Could you explain how the first inequality can be obtained?
Take $f(\tau) = \|u^T(\tau)w\|$ and $g(\tau) = 1$, and use $$ f(\tau)^2 = w^T u(\tau) u^T(\tau) w. $$