finite-dimensional

In the finite-dimensional case, the second condition above is implied by the first.

The statement is essentially the same as the finite-dimensional version.

Let A be an operator on a finite-dimensional inner product space.

In this case, B will be a finite-dimensional, real algebra.

When the vector space is not finite-dimensional, further distinctions are needed.

In the finite-dimensional case, there is a somewhat more explicit formulation.

This finite-dimensional problem is then implemented on a computer.

In particular, if either of or is finite-dimensional, these maps are isomorphisms.

In the case of finite-dimensional spaces, both theories perfectly match.

For simplicity, it will be assumed that all objects in the article are finite-dimensional.

Finite-dimensional division algebras over the real numbers are also very rare.

Let V be a finite-dimensional vector space over a field k.

All of these are again finite-dimensional vector spaces over k.

A vector space that has a finite basis is called finite-dimensional.

In fact, there are stronger statements than the absence of finite-dimensional representations.

For simplicity, the following assumes all relevant state spaces are finite-dimensional.

Note that a space is, by this definition, finite-dimensional.

Each module in a category O has finite-dimensional weight spaces.

This is true regardless of whether H is finite-dimensional or not.

Consider linear operators on a finite-dimensional vector space over a perfect field.

Jordan canonical form, where the finite-dimensional case is discussed in some detail.

If the vector space is a finite-dimensional real or complex one, all norms are equivalent.

They arise, for example, in the study of finite-dimensional modules over an algebra.

The most important distinction is between finite-dimensional representations and infinite-dimensional ones.

Additional algebraic structures can also be imposed in the finite-dimensional case.