Definition & Betydelse | Engelska ordet INFINITE-DIMENSIONAL

Definition av INFINITE-DIMENSIONAL

1. (matematik) oändligdimensionell

Exempel på hur man kan använda INFINITE-DIMENSIONAL i en mening

• Bra–ket notation, also called Dirac notation, is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional and infinite-dimensional case.
• A modern, abstract point of view contrasts large function spaces, which are infinite-dimensional and within which most functions are 'anonymous', with special functions picked out by properties such as symmetry, or relationship to harmonic analysis and group representations.
• The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces.
• The concept of normal matrices can be extended to normal operators on infinite-dimensional normed spaces and to normal elements in C*-algebras.
• Often these models are infinite-dimensional, rather than finite dimensional, as is parametric statistics.
• Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional Kac–Moody algebras.
• The simplification reduces the state space of the system to a finite dimension, and the partial differential equations (PDEs) of the continuous (infinite-dimensional) time and space model of the physical system into ordinary differential equations (ODEs) with a finite number of parameters.
• Every linear functional is a linear map and on every infinite-dimensional normed space, there exists some discontinuous linear functional.
• An infinite-dimensional CW complex can be constructed by repeating the above process countably many times.
• In physics it is sometimes said the conformal group is infinite-dimensional, but this is not quite correct as while the Lie algebra of local symmetries is infinite dimensional, these do not necessarily extend to a Lie group of well-defined global symmetries.
• With Bernard Maurey he resolved the "unconditional basic sequence problem" in 1992, showing that not every infinite-dimensional Banach space has an infinite-dimensional subspace that admits an unconditional Schauder basis.
• Nelson made contributions to the theory of infinite-dimensional group representations, the mathematical treatment of quantum field theory, the use of stochastic processes in quantum mechanics, and the reformulation of probability theory in terms of non-standard analysis.
• In an infinite-dimensional space V, as used in functional analysis, the flag idea generalises to a subspace nest, namely a collection of subspaces of V that is a total order for inclusion and which further is closed under arbitrary intersections and closed linear spans.
• Some natural questions are: under what circumstances does this formalism work, and for what operators L are expansions in series of other operators like this possible? Can any function f be expressed in terms of the eigenfunctions (are they a Schauder basis) and under what circumstances does a point spectrum or a continuous spectrum arise? How do the formalisms for infinite-dimensional spaces and finite-dimensional spaces differ, or do they differ? Can these ideas be extended to a broader class of spaces? Answering such questions is the realm of spectral theory and requires considerable background in functional analysis and matrix algebra.
• Indeed, introductory physics texts on quantum mechanics often gloss over mathematical technicalities that arise for continuous-valued observables and infinite-dimensional Hilbert spaces, such as the distinction between bounded and unbounded operators; questions of convergence (whether the limit of a sequence of Hilbert-space elements also belongs to the Hilbert space), exotic possibilities for sets of eigenvalues, like Cantor sets; and so forth.
• However, some partial results have been established: for instance, any self-adjoint operator on an infinite-dimensional real Hilbert space admits an AIHS, as does any strictly singular (or compact) operator acting on a real infinite-dimensional reflexive space.
• The technical difficulty of their work is that geodesics in their infinite-dimensional context may have low differentiability.
• In mathematics, a number of fixed-point theorems in infinite-dimensional spaces generalise the Brouwer fixed-point theorem.
• An example of a classifying space is that when G is cyclic of order two; then BG is real projective space of infinite dimension, corresponding to the observation that EG can be taken as the contractible space resulting from removing the origin in an infinite-dimensional Hilbert space, with G acting via v going to −v, and allowing for homotopy equivalence in choosing BG.
• But also, one can study the complex representations of the group G(k) when k is a finite field, or the infinite-dimensional unitary representations of a real reductive group, or the automorphic representations of an adelic algebraic group.

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