Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License
 adj. (of a vector space) having a basis consisting of a finite number of elements.
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Examples

All finitedimensional inner product spaces are complete, and I will restrict myself to these.

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).

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.

Every finitedimensional vector space is free, being generated by any choice of basis.

The essence of duality for finitedimensional vector spaces resides in its involutary nature along with the reversal of the linear transformations.

In the finitedimensional case, multiplication is realized as the usual matrix product.

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)).

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

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.

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