# there exist additive functions which are not linear

###### Proof.

Let $V$ be the infinite dimensional vector space^{} $\mathbb{R}$ over the
field $\mathbb{Q}$. Since $1$ and $\sqrt{2}$ are two independent vectors in $V$, we can extend the set $\{1,\sqrt{2}\}$ to a basis $E$ of $V$ (notice that here the axiom of choice^{} is used).

Now we consider a linear function^{} $f:V\to \mathbb{R}$ such that $f(1)=1$ while $f(e)=0$ for all $e\in E\setminus \{1\}$. This function is $\mathbb{Q}$-linear (i.e. it is additive on $\mathbb{R}$) but it is not $\mathbb{R}$-linear because $f(\sqrt{2})=0\ne \sqrt{2}f(1)$.
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Title | there exist additive functions which are not linear |
---|---|

Canonical name | ThereExistAdditiveFunctionsWhichAreNotLinear |

Date of creation | 2013-03-22 16:17:50 |

Last modified on | 2013-03-22 16:17:50 |

Owner | paolini (1187) |

Last modified by | paolini (1187) |

Numerical id | 5 |

Author | paolini (1187) |

Entry type | Example |

Classification | msc 15A04 |