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 (200)
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: High Entry average rating: No information on entry rating
subadditive (Definition)

Suppose $ V$ is a vector space (over a field), and $ f$ is a function $ f\colon V\to \mathbbmss{R}$. Then $ f$ is subadditive if

$\displaystyle f(x+y) \le f(x) + f(y), \quad x,y\in V. $

Examples

  1. Any linear function $ V\to \mathbbmss{R}$ is subadditive.
  2. If $ \Vert\cdot \Vert$ is a norm on $ V$, $ a\ge 0$, then
    $\displaystyle f(x) = a+\Vert x \Vert $
    is subadditive.

Properties

Suppose $ f$ is subadditive.
  1. If $ f$ is positively $ 1$-homogeneous, then $ f$ is convex.
  2. The sum of two subadditive functions is subadditive.



Anyone with an account can edit this entry. Please help improve it!

"subadditive" is owned by matte. [ full author list (2) ]
(view preamble)

View style:

See Also: linear transformation, seminorm, homogeneous function
Log in to rate this entry.
(view current ratings)

Cross-references: sum, convex, norm, function, field, vector space

This is version 4 of subadditive, born on 2005-04-26, modified 2006-10-02.
Object id is 6967, canonical name is SubLinear.
Accessed 1275 times total.

Classification:
AMS MSC: 46B20 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Geometry and structure of normed linear spaces)
Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

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