The Uniform Boundedness Principle

1985 ◽  
pp. 21-28
Author(s):  
Piotr Antosik ◽  
Charles Swartz
Keyword(s):  
Uniform Boundedness ◽  
Uniform Boundedness Principle

Banach Spaces

2021 ◽  
pp. 245-289
Author(s):  
Adel N. Boules
Keyword(s):  
Banach Spaces ◽  
Uniform Boundedness ◽  
Linear Transformations ◽  
Closed Graph Theorem ◽  
Complemented Subspaces ◽  
Good Account ◽  
Banach Theorem ◽  
Schauder Bases ◽  
Uniform Boundedness Principle ◽  
Adjoint Operators

The first four sections of this chapter form its core and include classical topics such as bounded linear transformations, the open mapping theorem, the closed graph theorem, the uniform boundedness principle, and the Hahn-Banach theorem. The chapter includes a good number of applications of the four fundamental theorems of functional analysis. Sections 6.5 and 6.6 provide a good account of the properties of the spectrum and adjoint operators on Banach spaces. They may be largely bypassed, since the treatment of the corresponding topics for operators on Hilbert spaces in chapter 7 is self-contained. The section on weak topologies is more advanced and may be omitted if a brief introduction is the goal. The chapter is enriched by such topics as the best polynomial approximation, the Hilbert cube, Gelfand’s theorem, Schauder bases, complemented subspaces, and the Banach-Alaoglu theorem.

The uniform boundedness principle for bornological vector spaces

1994 ◽  
Vol 62 (3) ◽  
pp. 270-277
Author(s):  
Claude-Alain Faure ◽  
Alfred Fr�licher
Keyword(s):  
Uniform Boundedness ◽  
Vector Spaces ◽  
Uniform Boundedness Principle

scholarly journals A generalisation of Mackey's theorem and the uniform boundedness principle

1989 ◽  
Vol 40 (1) ◽  
pp. 123-128 ◽  
Author(s):  
Charles Swartz
Keyword(s):  
Uniform Boundedness ◽  
Mackey Topology ◽  
Convex Topology ◽  
Uniform Boundedness Principle ◽  
Bounded Sets ◽  
Locally Convex Topology ◽  
Locally Convex

We construct a locally convex topology which is stronger than the Mackey topology but still has the same bounded sets as the Mackey topology. We use this topology to give a locally convex version of the Uniform Bouudedness Principle which is valid without any completeness or barrelledness assumptions.

scholarly journals The uniform boundedness principle for order bounded operators

1989 ◽  
Vol 12 (3) ◽  
pp. 487-492 ◽  
Author(s):  
Charles Swartz
Keyword(s):  
Uniform Boundedness ◽  
Linear Operators ◽  
Bounded Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Uniform Boundedness Principle ◽  
Steinhaus Theorem

Under appropriate hypotheses on the spaces, it is shown that a sequence of order bounded linear operators which is pointwise order bounded is uniformly order bounded on order bounded subsets. This result is used to establish a Banach-Steinhaus Theorem for order bounded operators.

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