On the Riesz-Fischer theorem

2004 ◽  
Vol 41 (4) ◽  
pp. 467-478
Author(s):  
J. Horváth
Keyword(s):  
Representation Theorem ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Riesz Representation Theorem ◽  
Riesz Representation

Are the two forms in which the theorem of the title is usually stated equivalent? We first summarize the three Comptes Rendus notes in which Frédéric Riesz published his results concerning L2, and then, in somewhat more detail, an article from 1910 which has been published only in Hungarian. Riesz deduces the two forms not from each other but both from the Fréchet—Riesz representation theorem. A theorem states that some of Riesz's results hold in the case of an abstract inner product space, and leads to maximal orthonormal systems which are not total. We conclude with a proof due to Ákos Császár which shows that a variant of Riesz's condition implies the Fischer form (i.e., completeness).

scholarly journals Inner Product Groups and Riesz Representation Theorem

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1946
Author(s):  
Alireza Pourmoslemi ◽  
Tahereh Nazari ◽  
Mehdi Salimi
Keyword(s):  
Representation Theorem ◽  
Abelian Groups ◽  
Inner Product ◽  
Riesz Representation Theorem ◽  
Product Group ◽  
Basic Properties ◽  
Inner Products ◽  
Normed Group ◽  
Riesz Representation ◽  
Torsion Free

In this paper, we introduce an inner product on abelian groups and, after investigating the basic properties of the inner product, we first show that each inner product group is a torsion-free abelian normed group. We give examples of such groups and describe the norms induced by such inner products. Among other results, Hilbert groups, midconvex and orthogonal subgroups are presented, and a Riesz representation theorem on divisible Hilbert groups is proved.

scholarly journals Fuzzy Inner Product Space: Literature Review and a New Approach

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 765
Author(s):  
Lorena Popa ◽  
Lavinia Sida
Keyword(s):  
Literature Review ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Comprehensive Overview ◽  
New Approach ◽  
Inner Product Spaces ◽  
Product Function ◽  
Suitable Definition

The aim of this paper is to provide a suitable definition for the concept of fuzzy inner product space. In order to achieve this, we firstly focused on various approaches from the already-existent literature. Due to the emergence of various studies on fuzzy inner product spaces, it is necessary to make a comprehensive overview of the published papers on the aforementioned subject in order to facilitate subsequent research. Then we considered another approach to the notion of fuzzy inner product starting from P. Majundar and S.K. Samanta’s definition. In fact, we changed their definition and we proved some new properties of the fuzzy inner product function. We also proved that this fuzzy inner product generates a fuzzy norm of the type Nădăban-Dzitac. Finally, some challenges are given.

On Closed Subsets of Root Systems

1994 ◽  
Vol 37 (3) ◽  
pp. 338-345 ◽  
Author(s):  
D. Ž. Doković ◽  
P. Check ◽  
J.-Y. Hée
Keyword(s):  
Weyl Group ◽  
Root System ◽  
Irreducible Component ◽  
Product Space ◽  
Root Systems ◽  
Inner Product Space ◽  
Inner Product ◽  
Closed Sets ◽  
Finite Dimensional ◽  
Real Inner Product Space

AbstractLet R be a root system (in the sense of Bourbaki) in a finite dimensional real inner product space V. A subset P ⊂ R is closed if α, β ∊ P and α + β ∊ R imply that α + β ∊ P. In this paper we shall classify, up to conjugacy by the Weyl group W of R, all closed sets P ⊂ R such that R\P is also closed. We also show that if θ:R —> R′ is a bijection between two root systems such that both θ and θ-1 preserve closed sets, and if R has at most one irreducible component of type A1, then θ is an isomorphism of root systems.

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.

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