I want to prove Cauchy–Schwarz' inequality, in Dirac notation, $\left<\psi\middle|\psi\right> \left<\phi\middle|\phi\right> \geq \left|\left<\psi\middle|\phi\right>\right|^2$, using the Lagrange multiplier method, minimizing $\left|\left<\psi\middle|\phi\right>\right|^2$ subject to the constraint $\left<\phi\middle|\phi\right> - c = 0$, where $c$ is a constant.
I'm completely new to Lagrange multipliers, and the Fréchet derivative etc., and have tried to consult https://en.wikipedia.org/wiki/Lagrange_multipliers_on_Banach_spaces, but am still quite confused, conceptually.
This is my sketchy thinking thus far (trying to follow the Wikipedia exposition, adapted to my problem):
We have a Banach space $B_\phi$. We then have $f = \left|\left<\psi\middle|\phi\right>\right|^2 : B_\phi \to \mathbf C$, which we want to minimize. The constraint is given by $g = \left<\phi\middle|\phi\right> - c : B_\phi \to \mathbf C$, which is set to zero. The Wikipedia article goes on to suppose that "$u_0$" (would "$ \left|\phi_0\right>$" be a logical label in my case?) is a constrained extremum of $f$, i.e. an extremum of $f$ on $g^{-1}(0) = \big\{\left|\phi\right> \in B_\phi$ $|$ $g(\left|\phi\right>) = 0 \in \mathbf C \big\} \subseteq B_\phi$. The problem is then formulated as $$Df(u_0) = \lambda \circ Dg(u_0)$$ where $\lambda$ is the Lagrange multiplier, and D the Fréchet derivative. Is it a complete misconception if I write this as (given $f$ and $g$ above, and my assumption that $u_0 = \left|\phi_0\right>$) $$D \left|\left<\psi|\phi_0\right>\right|^2 = \lambda \circ D\big(\left<\phi_0|\phi_0\right> -c\big)$$?
My main questions at the moment are:
- What are the conceptual errors above? (I guess there are plenty)
- How do I evaluate the Fréchet derivative, e.g. $D \left|\left<\psi|\phi_0\right>\right|^2$?
Thanks in advance!