---
tag: 018W3V3TCT
kind: definition
name: Finite-dimensional vector space
msc: [15A03]
defines: 0ZHWKQMVX5
uses: [0BJKVF90Z6]
sources:
  - url: https://en.wikipedia.org/wiki/Dimension_(vector_space)
    locator: lead section
---

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
The space $V$ is *finite-dimensional* when it has a finite [[spanning set|088RYKAWK9]].
Otherwise $V$ is *infinite-dimensional*.

## Remarks

A finite-dimensional space has a finite basis, because [[every spanning set contains a basis|06MRNP4BNT]].
The spaces $F^n$ and $M_{m \times n}(F)$ are finite-dimensional.
The polynomial ring $F[x]$ is infinite-dimensional: finitely many polynomials have a bounded degree, so they do not span $F[x]$.
Each [[subspace|00YC1D652K]] of a finite-dimensional space is finite-dimensional.

## Explanations

### L1 human

A space is finite-dimensional when finitely many vectors are enough to build every vector.
The plane needs two vectors, and ordinary space needs three.
The space of all polynomials needs infinitely many, because the degrees have no bound.