finite-dimensional love



from Wiktionary, Creative Commons Attribution/Share-Alike License.

  • adjective mathematics (of a vector space) having a basis consisting of a finite number of elements.


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  • All finite-dimensional inner product spaces are complete, and I will restrict myself to these.

    Quantum Mechanics Ismael, Jenann 2009

  • If the quantum state evolves in a finite-dimensional Hilbert space, it evolves ergodically through a torus of phases, and will exhibit all of the usual problems of Boltzmann brains and the like (as Dyson, Kleban, and Susskind have emphasized).

    What if Time Really Exists? Sean 2008

  • Every finite-dimensional vector space is free, being generated by any choice of basis.

    Algebra Pratt, Vaughan 2007

  • The essence of duality for finite-dimensional vector spaces resides in its involutary nature along with the reversal of the linear transformations.

    Algebra Pratt, Vaughan 2007

  • However, relying on the (near) ubiquity of decoherence in the macroscopic realm, Bacciagaluppi and Hemmo show that when the apparatus is considered as a finite-dimensional system

    Modal Interpretations of Quantum Mechanics Dickson, Michael 2007

  • In the finite-dimensional case, multiplication is realized as the usual matrix product.

    Algebra Pratt, Vaughan 2007

  • It would seem at first sight that a lattice formulation could never reproduce this, since in finite volume (e.g. on the 4-torus) the space of lattice spinor fields is finite-dimensional but to have a non-trivial index theory the operators generally must be acting on an infinite-dimensional vectorspace.

    String Theory is Losing the Public Debate Sean 2007

  • This result applies more generally to other cases where a macroscopic system (not idealized as finite-dimensional) experiences decoherence due to interaction with its environment (see Donald (1998)).

    Modal Interpretations of Quantum Mechanics Dickson, Michael 2007

  • In classical Hamilton-Jacobi theory we also have this equation for the velocity, but there the Hamilton-Jacobi function S can be entirely eliminated and the description in terms of S simplified and reduced to a finite-dimensional description, with basic variables the positions and the (unconstrained) momenta of all the particles, given by Hamilton's or Newton's equations.

    Bohmian Mechanics Goldstein, Sheldon 2006

  • (If the domain is both compact and discrete, then it is finite, and on a finite-dimensional space all norms are equivalent.)

    What's new Terence Tao 2010


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