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 (194)
Orphanage
Unclass'd
Unproven (498)
Corrections (25)
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
continuous linear mapping (Definition)

If $ (V_1,\Vert\cdot\Vert _1)$ and $ (V_2,\Vert\cdot\Vert _2)$ are normed vector spaces, a linear mapping $ T:V_1\rightarrow V_2$ is continuous if it is continuous in the metric induced by the norms.

If there is a nonnegative constant $ c$ such that $ \Vert T(x)\Vert _2\leq c\Vert x\Vert _1$ for each $ x\in V_1$, we say that $ T$ is bounded. This should not be confused with the usual terminology referring to a bounded function as one that has bounded range. In fact, bounded linear mappings usually have unbounded ranges.

The expression bounded linear mapping is often used in functional analysis to refer to continuous linear mappings as well. This is because the two definitions are equivalent:

If $ T$ is bounded, then $ \Vert T(x)-T(y)\Vert _2 = \Vert T(x-y)\Vert _2 \leq c\Vert x-y\Vert _1$, so $ T$ is a Lipschitz function. Now suppose $ T$ is continuous. Then there exists $ r>0$ such that $ \Vert T(x)\Vert _2 \leq 1$ when $ \Vert x\Vert _1\leq r$. For any $ x\in V_1$, we then have

$\displaystyle \frac{r}{\Vert x\Vert _1}\Vert T(x)\Vert _2 = \Vert T\left(\frac{r}{\Vert x\Vert _1}x\right)\Vert _2 \leq 1,$
hence $ \Vert T(x)\Vert _2\leq r\Vert x\Vert _1$; so $ T$ is bounded.

It can be shown that a linear mapping between two topological vector spaces is continuous if and only if it is continuous at 0 [1].

Bibliography

1
W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.



"continuous linear mapping" is owned by Koro.
(view preamble)

View style:

Other names:  bounded linear mapping
Also defines:  bounded linear transform, bounded linear operator

Attachments:
bounded linear extension of an operator (Theorem) by asteroid
Log in to rate this entry.
(view current ratings)

Cross-references: topological vector spaces, Lipschitz function, equivalent, definitions, functional analysis, expression, unbounded, range, bounded, bounded function, metric induced by the norms, continuous, linear mapping, normed vector spaces
There are 21 references to this entry.

This is version 4 of continuous linear mapping, born on 2002-12-13, modified 2003-08-04.
Object id is 3741, canonical name is ContinuousLinearMapping.
Accessed 8407 times total.

Classification:
AMS MSC: 46B99 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Miscellaneous)
Pending Errata and Addenda
None.
[ View all 2 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

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