- 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
- Outer product in Hilbert space - Physics Forums
But it comes closest to the PHYSICAL reason of why H (x) H' is the hilbert space of a combined system Indeed, a basis of the hilbert space of a system gives you normally an exhaustive list of mutually exclusive states of that system, and the whole statespace is then obtained by a superposition of all of these states
- 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
- 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 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
- 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
- Dimensions of Hilbert Spaces confusion - Physics Forums
If I understand it, Hilbert spaces can be finite (e g , for spin of a particle), countably infinite (e g , for a particle moving in space), or uncountably infinite (i e , non-separable, e g , QED) I am wondering about variations on this latter The easiest uncountable to imagine is the
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