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## Main Question or Discussion Point

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 (although the idea is perfectly clear in simpler cases like e.g. ##f : \mathbf R^2 \to \mathbf R##), 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 let ##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:

1. What are the conceptual errors above? (I guess there are plenty)

2. How do I evaluate the Fréchet derivative, e.g. ##D \left|\left<\psi|\phi_0\right>\right|^2##?

Thanks in advance!

I'm completely new to Lagrange multipliers (although the idea is perfectly clear in simpler cases like e.g. ##f : \mathbf R^2 \to \mathbf R##), 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 let ##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:

1. What are the conceptual errors above? (I guess there are plenty)

2. How do I evaluate the Fréchet derivative, e.g. ##D \left|\left<\psi|\phi_0\right>\right|^2##?

Thanks in advance!