scholarly journals Banach-Steinhaus Theorem for the Space P of All Primitives of Henstock-Kurzweil Integrable Functions

10.53733/114 ◽  
2021 ◽  
Vol 51 ◽  
pp. 79-83
Author(s):  
Wee Leng Ng
Keyword(s):  
Representation Theorem ◽  
Uniform Norm ◽  
Riesz Representation Theorem ◽  
Bounded Interval ◽  
Integrable Functions ◽  
Banach Theorem ◽  
Riesz Representation ◽  
Steinhaus Theorem

In this paper, it is shown how the Banach-Steinhaus theorem for the space P of all primitives of Henstock-Kurzweil integrable functions on a closed bounded interval, equipped with the uniform norm, can follow from the Banach-Steinhaus theorem for the Denjoy space by applying the classical Hahn-Banach theorem and Riesz representation theorem.   

Another ‘short’ proof of the Riesz representation theorem

1986 ◽  
Vol 99 (2) ◽  
pp. 261-262 ◽  
Author(s):  
D. J. H. Garling
Keyword(s):  
Representation Theorem ◽  
Short Proof ◽  
Discrete Space ◽  
Riesz Representation Theorem ◽  
Banach Theorem ◽  
Riesz Representation ◽  
Synthetic Proof

1. Introduction. In [2], a short synthetic proof of the Riesz representation theorem was given; this used the Hahn-Banach theorem, the Stone-Čech compactification of a discrete space and the Caratheodory extension procedure for measures. In this note, we show how the theorem can be proved using ultrapowers in place of the Stone-Čech compactification. We also describe how the proof can be expressed in a non-standard way (a rather different non-standard proof has been given by Loeb [4]).

ON THE RIESZ REPRESENTATION THEOREM IN TOPOLOGICAL VECTOR SPACES

2000 ◽  
Vol 36 (3-4) ◽  
pp. 347-352
Author(s):  
M. A. Alghamdi ◽  
L. A. Khan ◽  
H. A. S. Abujabal
Keyword(s):  
Representation Theorem ◽  
Type Theorem ◽  
Representation Type ◽  
Vector Spaces ◽  
Riesz Representation Theorem ◽  
Topological Vector Spaces ◽  
Riesz Representation

I this paper we establish a Riesz representation type theorem which characterizes the dual of the space C rc (X,E)endowed with the countable-ope topologyi the case of E ot ecessarilya locallyconvex TVS.

scholarly journals A note on the Riesz representation theorem

1963 ◽  
Vol 14 (2) ◽  
pp. 354-354 ◽  
Author(s):  
Don H. Tucker
Keyword(s):  
Representation Theorem ◽  
Riesz Representation Theorem ◽  
Riesz Representation

The Riesz Representation Theorem and Integral Norms

1968 ◽  
Vol s1-43 (1) ◽  
pp. 177-182 ◽  
Author(s):  
B. L. D. Thorp
Keyword(s):  
Representation Theorem ◽  
Riesz Representation Theorem ◽  
Riesz Representation

Yet another short proof of the Riesz representation theorem

1989 ◽  
Vol 105 (1) ◽  
pp. 139-140 ◽  
Author(s):  
David Ross
Keyword(s):  
Representation Theorem ◽  
Short Proof ◽  
Riesz Representation Theorem ◽  
Riesz Theorem ◽  
Riesz Representation

F. Riesz's ‘Representation Theorem’ has been proved by methods classical [11, 12], category-theoretic [7], and functional-analytic [2, 9]. (Garling's remarkable proofs [5, 6] owe their brevity to the combined strength of these and other methods.) These proofs often reveal a connection between the Riesz theorem and some unexpected area of mathematics.

A nonstandard proof of the Riesz representation theorem for weakly compact operators on C(Ω)

1989 ◽  
Vol 105 (1) ◽  
pp. 141-145
Author(s):  
Yeneng Sun
Keyword(s):  
Representation Theorem ◽  
Compact Operators ◽  
Riesz Representation Theorem ◽  
Linear Functionals ◽  
Weakly Compact ◽  
Vector Measures ◽  
Riesz Representation ◽  
Weakly Compact Operators ◽  
Measure Technique

AbstractAn easy way to construct the representing vector measures of weakly compact operators on C(Ω) is given by using the Loeb measure technique. This construction is not based on the Riesz representation theorem for linear functionals; thus we have a uniform way to treat the scalar and vector cases. Also the star finite representations of regular vector measures follow from the proof.

Extension of the riesz representation theorem and solution of the Maxwell-Boltzmann functional equation

1970 ◽  
Vol 5 (1) ◽  
pp. 125-126
Author(s):  
E. D. Cashwell ◽  
C. J. Everett
Keyword(s):  
Functional Equation ◽  
Representation Theorem ◽  
Riesz Representation Theorem ◽  
Riesz Representation
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