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- Learn the Basics of Hilbert Spaces and Their Relatives: Definitions
Since Pre-Hilbert spaces are a class of spaces, whereas the rational numbers are a specific field, Pre-Hilbert spaces merely do not need to be incomplete, they may as well be complete, in which case we call them Hilbert spaces I follow the convention, that Pre-Hilbert spaces contain Hilbert spaces, and not that they are complementary
- What Distinguishes Hilbert Spaces from Euclidean Spaces? - Physics Forums
Hilbert spaces are not necessarily infinite dimensional, I don't know where you heard that Euclidean space IS a Hilbert space, in any dimension or even infinite dimensional A Hilbert space is a complete inner product space An inner product space is a vector space with an inner product defined on it
- Where does the Einstein-Hilbert action come from? - Physics Forums
Einstein, to favor the simplest form of the stress-energy equation among many candidates, obtained the same result as Hilbert--perhaps motivated, in part, by the conclusion of Hilbert, I would suspect Hilbert, on the other hand, was very missive in taking credit, and insisted that Einstein deserved the credit The action is the invention of
- Why is Hilbert not the last universalist? - Physics Forums
Here is the section on Hilbert's work on Physics from the Wikipedia article "David Hilbert" At the end of the article, the Book "Mathematical Methods of Physics" by Courant and Hilbert is mentioned It is an attempt to bring Mathematics and Physics together as is explicitly stated in the introduction "Physics
- Difference between hilbert space,vector space and manifold?
A Hilbert space is a vector space with a defined inner product This means that in addition to all the properties of a vector space, I can additionally take any two vectors and assign to them a positive-definite real number This assignment has to satisfy some additional properties It has to be 0 only if one of the vectors I give it is 0
- Why are Hilbert spaces used in quantum mechanics? - Physics Forums
Note that 'Hilbert space' is a pretty general concept, every vector space with an inner product (and no 'missing points' e g like real numbers but not only the rational ones) is a hilbert space Classical phase space is also a Hilbert space in this sense where positions and momenta constitute the most useful basis vectors
- Derivation of the Einstein-Hilbert Action - Physics Forums
Derivation of the Einstein-Hilbert Action Abstract Most people justify the form of the E-H action by saying that it is the simplest scalar possible But simplicity, one can argue, is a somewhat subjective and ill-defined criterion Also, simplicity does not shed light on the axiomatic structure of general relativity
- Hilbert spaces and quantum operators being infinite dimensional matrices
If you want to study the Hilbert space approach (as opposed to the rigged Hilbert space approach), then I recommend the beautiful "Quantum Theory for Mathematicians" by Brian Hall over Reed and Simon, even though I like Reed and Simon Hall is a really wonderful book, but it has "the basics of L^2 spaces and Hilbert spaces" as prerequisites
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