---
tag: 0C13E623WD
kind: statement
role: theorem
name: Linear maps on finite-dimensional spaces are continuous
msc: [47B01]
uses: [0F3K4C7XYW, 03MC70ZRNR, 0S3SMRH0Z5]
sources:
  - url: https://en.wikipedia.org/wiki/Bounded_operator
    locator: Examples
---

## Statement

Let $V$ be a finite-dimensional normed space and let $W$ be a normed space, both over $\mathbb{R}$ or over $\mathbb{C}$.
Then each linear map $T \colon V \to W$ is [[continuous|0S3SMRH0Z5]].

## Remarks

Proof: choose a basis $e_1, \ldots, e_n$ of $V$. For $x = \sum_i a_i e_i$, the triangle inequality gives $\|Tx\|_W \le \left( \sum_i \|T e_i\|_W \right) \max_i |a_i|$. By [[equivalence of norms in finite dimension|03MC70ZRNR]], $\max_i |a_i| \le c \|x\|_V$ for some $c$. Thus $T$ is bounded, and so [[continuous|0F3K4C7XYW]].
The finite dimension of $W$ is not needed, only that of $V$.
For example, each matrix gives a continuous map $\mathbb{R}^n \to \mathbb{R}^m$.

## Explanations

### L1 human

In finite dimensions, linear maps are automatically continuous, for each choice of norms.
Problems with continuity only appear in infinite dimensions.