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 (231)
Orphanage
Unclass'd
Unproven (514)
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
[parent] pointwise limit of bounded operators is bounded (Corollary)

The following result is a corollary of the principle of uniform boundedness.

Theorem - Let $ X$ be a Banach space and $ Y$ a normed vector space. Let $ (T_n) \in B(X,Y)$ be a sequence of bounded operators from $ X$ to $ Y$. If $ (T_nx)$ converges for every $ x \in X$, then the operator

$ T:X \longrightarrow Y$

$\displaystyle Tx = \lim_{n \rightarrow \infty} T_n x $
is linear and bounded. Moreover, the sequence $ (\Vert T_n\Vert)$ is bounded.

Proof : It is clear that the operator $ T$ is linear.

For each $ x \in X$ we have $ \displaystyle \;\sup_n \Vert T_nx\Vert < \infty\;$ since $ (T_nx)$ is convergent. By the principle of uniform boundedness we must also have $ \displaystyle M := \sup_n \Vert T_n\Vert < \infty$.

Then for each $ x \in X$ we have

$\displaystyle \Vert Tx\Vert = \lim_{n \rightarrow \infty} \Vert T_nx\Vert \leq M\Vert x\Vert $

which means that $ T$ is bounded. $ \square$



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

"pointwise limit of bounded operators is bounded" is owned by asteroid.
(view preamble | get metadata)

View style:


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

Cross-references: clear, proof, operator, converges, bounded operators, sequence, normed vector space, Banach space, Principle of Uniform Boundedness
There is 1 reference to this entry.

This is version 1 of pointwise limit of bounded operators is bounded, born on 2007-09-13.
Object id is 9933, canonical name is PointwiseLimitOfBoundedOperatorsIsBounded.
Accessed 622 times total.

Classification:
AMS MSC: 46B99 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Miscellaneous)
 47A05 (Operator theory :: General theory of linear operators :: General )
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

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