Lagrange multipliers on Banach spaces (in Dirac notation)

In summary, the conversation discusses the use of the Lagrange multiplier method to prove Cauchy-Schwarz' inequality in Dirac notation. The problem involves 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. The main questions are about conceptual errors and evaluating the Fréchet derivative.
  • #1
Rabindranath
10
1
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!
 
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  • #2
Rabindranath said:
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.
 
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Likes Rabindranath
  • #3
Krylov said:
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.
 

Related to Lagrange multipliers on Banach spaces (in Dirac notation)

1. What are Lagrange multipliers on Banach spaces?

Lagrange multipliers on Banach spaces are a mathematical tool used to find the extrema (maximum or minimum) of a function subject to constraints. In Dirac notation, this involves finding the stationary points of a functional with the help of a Lagrange multiplier.

2. How are Lagrange multipliers used in Banach spaces?

In Banach spaces, Lagrange multipliers are used to optimize a functional by finding the critical points of the function. This involves setting up an augmented functional with a Lagrange multiplier term and solving for the stationary points using the calculus of variations.

3. What is the role of Dirac notation in Lagrange multipliers on Banach spaces?

Dirac notation, also known as bra-ket notation, is a mathematical notation used to represent vectors and linear operators in quantum mechanics. In the context of Lagrange multipliers on Banach spaces, Dirac notation is used to represent the functional and its constraints, making it easier to manipulate and solve for the critical points.

4. Can Lagrange multipliers be used on any Banach space?

Yes, Lagrange multipliers can be used on any Banach space, as long as the functional and constraints are defined on that space. However, the complexity of the space and the functional may affect the difficulty of solving for the critical points using Lagrange multipliers.

5. What are some applications of Lagrange multipliers on Banach spaces?

Lagrange multipliers on Banach spaces have numerous applications in math, physics, and engineering. They are commonly used in optimization problems, such as finding the shortest path between two points on a curved surface. They are also used in variational methods, such as the Euler-Lagrange equation, to solve differential equations. Additionally, Lagrange multipliers have applications in quantum mechanics, economics, and game theory.

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