A finite-dimensional vector space is a vector space where some actual list (which remember, has finite length) of vectors spans the space.

An infinite-demensional vector space is a vector space that’s not a finite-dimensional vector space.

## additional information

### every finite-dimensional vector space has a basis

Begin with a spanning list in the finite-dimensional vector space you are working with. Apply the fact that all spanning lists contains a basis of which you are spanning. Therefore, some elements of that list form a basis of the finite-dimensional vector space you are working with. \(\blacksquare\)