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
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
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
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!
Isomorphism Isometry: Hilbert Spaces - Physics Forums Hi, I am wondering if all isomorphisms between hilbert spaces are also isometries, that is, norm preserving In another sense, since all same dimensional hilbert spaces are isomorphic, are they all related by isometries also? Thank you,
Where does the Einstein-Hilbert action come from? The Hilbert action comes from postulating that gravity comes from making the metric dynamical, and that the dynamical equations come from an action, which is a scalar There are more complex terms consistent with this idea, and the Hilbert action is only the simplest
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
Orthogonal complement of the orthogonal complement - Physics Forums Main Points Raised One participant presents a proof showing that if M is a linear subspace of a Hilbert space H, then M ⊆ M ⊥⊥, suggesting that the topological closure of M is M ⊥⊥ Another participant argues that the inclusion M ― ⊂ (M ⊥) ⊥ holds, emphasizing that orthogonal complements are closed linear subspaces