Lagrange multipliers on Banach spaces (in Dirac notation)

  • #1

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!
 

Answers and Replies

  • #2
S.G. Janssens
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2. How do I evaluate the Fréchet derivative, e.g. ##D \left|\left<\psi|\phi_0\right>\right|^2##?
As an example, suppose that you work in the complex Hilbert space ##H = \mathbb{C}## and let ##z_0 \in H## be fixed. If I read you correctly, you would be interested in differentiability of ##z \mapsto \left|\left<z |z_0 \right>\right|^2##. However, unless ##z_0 = 0## (which makes everything trivial), this map is not differentiable except at ##z = 0##. (You can check this using the Cauchy-Riemann equations.)

For the case of a real Hilbert space, it would be different. It is my impression that - since complex differentiability is such a strong requirement - Fréchet derivatives of operators and functionals are usually discussed in the context of real normed spaces, although I think that the basic definitions work fine in either case.
 
  • #3
As an example, suppose that you work in the complex Hilbert space ##H = \mathbb{C}## and let ##z_0 \in H## be fixed. If I read you correctly, you would be interested in differentiability of ##z \mapsto \left|\left<z |z_0 \right>\right|^2##.
Rather ##z \mapsto \left|\left<z_0 |z \right>\right|^2## if I use your example. That is, going back to my example again, for some arbitrarily chosen fixed element ##\left|\psi\right>## in the Banach space ##B_\phi##, I'm interested in the map ## \left|\phi\right> \mapsto \left|\left<\psi\middle|\phi\right>\right|^2 ##. This is what I wanted to minimize, by means of the Lagrange multiplier method (with constraint ##g = \left<\phi\middle|\phi\right> - c = 0 ##). What I called ##\left|\phi_0\right>## was meant as "the element that minimizes ## \left|\phi\right> \mapsto \left|\left<\psi\middle|\phi\right>\right|^2 ## given the constraint".

Thanks for your reply anyway! I will look into it deeper when I have more time.
 

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