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Learn the Basics of Hilbert Spaces and Their Relatives: Definitions Hilbert spaces are at first real or complex vector spaces, or are Hilbert spaces So all the theorems and definitions of linear algebra apply to the finite-dimensional ones and many to the infinite-dimensional ones, and we start at known ground
What Distinguishes Hilbert Spaces from Euclidean Spaces? 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
Hilbert Space Interpretation of Fourier Transform - Physics Forums The discussion revolves around the interpretation of the Fourier transform in the context of Hilbert spaces, particularly focusing on the relationship between Fourier series and continuous sets of eigenvalues Participants explore the mathematical foundations and implications of these concepts, especially as they relate to quantum mechanics and electrical engineering One participant notes
Why are Hilbert spaces used in quantum mechanics? Some participants suggest that Hilbert spaces are used in quantum mechanics due to the non-commutative nature of observables and the requirement for operators to act on states Others argue that Hilbert spaces are a general concept that includes classical phase spaces, which can also be considered Hilbert spaces under certain conditions
Why is Hilbert not the last universalist? • Physics Forums The discussion revolves around the characterization of mathematicians Hilbert and Poincaré as universalists, specifically questioning why Hilbert is not considered the last universalist despite his extensive knowledge in mathematics Participants explore various branches of mathematics, historical context, and the implications of their approaches to the discipline Some participants argue
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
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
Banach Space that is NOT Hilbert - Physics Forums I know that all Hilbert spaces are Banach spaces, and that the converse is not true, but I've been unable to come up with a (hopefully simple!) example of a Banach space that is not also a Hilbert space Any help would be appreciated!
Constructing Unitary Matrices for Rotations in Hilbert Space The discussion revolves around the construction of unitary matrices for rotations in Hilbert space, particularly focusing on their application to complex vector spaces Participants explore the differences between real and complex rotations, the nature of unitary operators, and the implications for manipulating complex vectors One participant inquires about constructing unitary matrices to