The Riesz representation theorem on infinite dimensional spaces

2003 ◽  
pp. 163-177
Author(s):  
Yuh-Jia Lee ◽  
Chen-Yuh Shih
Keyword(s):  
Representation Theorem ◽  
Riesz Representation Theorem ◽  
Infinite Dimensional ◽  
Riesz Representation

THE RIESZ REPRESENTATION THEOREM ON INFINITE DIMENSIONAL SPACES AND ITS APPLICATIONS

2002 ◽  
Vol 05 (01) ◽  
pp. 41-59
Author(s):  
YUH-JIA LEE ◽  
CHEN-YUH SHIH
Keyword(s):  
Representation Theorem ◽  
Borel Measure ◽  
Integrability Condition ◽  
Probabilistic Approach ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Nuclear Space ◽  
Riesz Representation Theorem ◽  
Infinite Dimensional ◽  
Riesz Representation

Let H be a real separable Hilbert space and let E⊂H be a nuclear space with the chain {Em: m=1,2,…} of Hilbert spaces such that E = ∩m∈ℕEm. Let E* and E-m denote the dual spaces of E and Em, respectively. For γ > 0, let [Formula: see text] be the collection of complex-valued continuous functions f defined on E* such that [Formula: see text] is finite for every m. Then [Formula: see text] is a complete countably normed space equipping with the family {‖·‖m,γ : m = 1,2,…} of norms. Using a probabilistic approach, it is shown that every continuous linear functional T on [Formula: see text] can be represented uniquely by a complex Borel measure νT satisfying certain exponential integrability condition. The results generalize an infinite dimensional Riesz representation theorem given previously by the first author for the case γ = 2. As applications, we establish a Weierstrass approximation theorem on E* for γ≥1 and show that the space [Formula: see text] spanned by the class { exp [i(x,ξ)] : ξ ∈ E} is dense in [Formula: see text] for γ>0. In the course of the proof we give sufficient conditions for a function space on which every positive functional can be represented by a Borel measure on E*.

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.

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.

Sign in / Sign up

Export Citation Format

Share Document