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Greg Bernhardt submitted a new PF Insights post
Hilbert Spaces and Their Relatives
Continue reading the Original PF Insights Post.
Hilbert Spaces and Their Relatives
Continue reading the Original PF Insights Post.
Hilbert Spaces and Their Relatives - Comments
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In summary, Greg Bernhardt has submitted a new PF Insights post titled "Hilbert Spaces and Their Relatives." The post discusses the importance of understanding Hilbert Spaces in quantum mechanics, as well as their applications in other areas. The concept of Rigged Hilbert Spaces is also mentioned, as well as the Fréchet-Riesz Representation Theorem and its proof in Ballentine's book. The link to Rigged Hilbert Spaces is provided for those who are unfamiliar with it, and it is noted that the test space must be dense.
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Mark44
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Nice job, fresh!
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bhobba
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Yes indeed a very nice job. Very important those that wish to progress in QM understand it, plus of course it has many other applications as well. When I learned it at uni my teacher said he could spend a 2 semester course on just the applications alone and still just scratch the surface. Thanks for being careful and mentioning its only isomorphic to its continuous dual - not its dual - I keep forgetting that one when explaining it to others which I do via the concept of Rigged Hilbert Spaces. If I remember correct its the same as demanding its bounded. Nice for people reading Ballentine QM - A Modern Approach. He gives his own proof of the Fréchet-Riesz Representation Theorem (page 10) but as he says it ignores convergence issues - in other words its wrong - but I will let others sort that one out (he is not careful with some manipulations he does on infinite series). I remember when first reading Ballentine all those years ago I thought naughty, naughty.
For those that do not know the link to Rigged Hilbert Spaces see:
https://www.univie.ac.at/physikwiki/images/4/43/Handout_HS.pdf
As I said I keep forgetting the continuous bit when I explain it - the above corrects it. Oh and the test space must be dense as well. Damn I am getting sloppy in my old age
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Thanks
Bill
For those that do not know the link to Rigged Hilbert Spaces see:
https://www.univie.ac.at/physikwiki/images/4/43/Handout_HS.pdf
As I said I keep forgetting the continuous bit when I explain it - the above corrects it. Oh and the test space must be dense as well. Damn I am getting sloppy in my old age
.Thanks
Bill
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1. What is a Hilbert space?
A Hilbert space is a mathematical concept that represents an infinite-dimensional vector space with a specific inner product. It is named after David Hilbert, a German mathematician, and is commonly used in functional analysis and quantum mechanics.
2. What are the key properties of a Hilbert space?
There are several key properties of a Hilbert space, including completeness, orthogonality, and the existence of an inner product. Completeness means that every Cauchy sequence in a Hilbert space converges to a point within the space. Orthogonality refers to the perpendicularity of vectors in the space, while the inner product is a mathematical operation that measures the angle between two vectors.
3. How are Hilbert spaces used in physics?
Hilbert spaces play a crucial role in quantum mechanics, where they are used to represent the state of a quantum system. They provide a mathematical framework for describing the probabilities of different outcomes of a measurement and for calculating the evolution of a system over time.
4. What are some examples of Hilbert spaces?
Some common examples of Hilbert spaces include the space of square-integrable functions, the space of square-summable sequences, and the space of continuous functions on a compact interval. These spaces have different structures and properties, but they all share the key characteristics of a Hilbert space.
5. What are the applications of Hilbert spaces in mathematics?
Hilbert spaces have many applications in mathematics, including functional analysis, signal processing, and optimization. They are also used in areas such as partial differential equations, statistical learning, and control theory. Hilbert spaces provide a powerful mathematical tool for solving complex problems in various fields.
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