Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 adjective mathematics (of a
vector space ) having a basis consisting of a finitenumber ofelements .
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Examples

All finitedimensional inner product spaces are complete, and I will restrict myself to these.
Quantum Mechanics Ismael, Jenann 2009

If the quantum state evolves in a finitedimensional 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 finitedimensional vector space is free, being generated by any choice of basis.
Algebra Pratt, Vaughan 2007

The essence of duality for finitedimensional 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 finitedimensional system
Modal Interpretations of Quantum Mechanics Dickson, Michael 2007

In the finitedimensional 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 4torus) the space of lattice spinor fields is finitedimensional but to have a nontrivial index theory the operators generally must be acting on an infinitedimensional vectorspace.

This result applies more generally to other cases where a macroscopic system (not idealized as finitedimensional) experiences decoherence due to interaction with its environment (see Donald (1998)).
Modal Interpretations of Quantum Mechanics Dickson, Michael 2007

In classical HamiltonJacobi theory we also have this equation for the velocity, but there the HamiltonJacobi function S can be entirely eliminated and the description in terms of S simplified and reduced to a finitedimensional 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 finitedimensional space all norms are equivalent.)
What's new Terence Tao 2010
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