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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
Understanding Spin States in Hilbert Space • Physics Forums Participants clarify that the Hilbert space of electron spin is a complex space and does not directly correspond to 3D physical space One participant seeks recommendations for texts that delve into the mathematics of Hilbert spaces, indicating a desire for deeper understanding beyond their current resources
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
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
Canonical Commutation Relations in finite dimensional Hilbert Space . . . The discussion revolves around the possibility of realizing the canonical commutation relations (CCR) in a finite-dimensional Hilbert space Participants explore the implications of these relations, particularly focusing on the trace operation and its consequences in both finite and infinite dimensions Some participants suggest that attempting to realize the CCR in a finite-dimensional
The History and Importance of the Riemann Hypothesis David Hilbert and Pólya György had already noticed that the Riemann hypothesis would follow if the zeros were eigenvalues of an operator where is a Hermitian (i e self-adjoint) operator, which therefore has only real eigenvalues, similar to the Hamiltonian operators in quantum mechanics
Has Anyone Ever Finished Reading Morse Feshbach and Courant Hilbert . . . The discussion revolves around the experiences and opinions of participants regarding the reading of mathematical and theoretical physics textbooks, specifically Morse Feshbach and Courant Hilbert Participants reflect on their reading habits, the challenges of understanding complex material, and the utility of these texts in academic and practical contexts Some participants express that
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