Definition of "finite-dimensional" []

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

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