PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (204)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Medium Entry average rating: No information on entry rating
[parent] there exist additive functions which are not linear (Example)
Example 1   There exists a function $ f\colon \mathbb{R}\to\mathbb{R}$ which is additive but 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\colon 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\neq \sqrt 2 f(1)$. $ \qedsymbol$



"there exist additive functions which are not linear" is owned by paolini.
(view preamble)

View style:


This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: axiom of choice, basis, vectors, independent, field, vector space, infinite dimensional, additive, function
There are 2 references to this entry.

This is version 2 of there exist additive functions which are not linear, born on 2006-10-04, modified 2006-10-05.
Object id is 8417, canonical name is ThereExistAdditiveFunctionsWhichAreNotLinear.
Accessed 990 times total.

Classification:
AMS MSC: 15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations)
Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add example | add (any)