Inner Product Spaces and Hilbert Spaces

2015 ◽  
pp. 213-225
Author(s):  
Philippe Blanchard ◽  
Erwin Brüning
Keyword(s):  
Hilbert Spaces ◽  
Inner Product ◽  
Product Spaces ◽  
Inner Product Spaces

scholarly journals Some inequalities for the norm and the numerical radius of linear operators in Hilbert spaces

2008 ◽  
Vol 39 (1) ◽  
pp. 1-7 ◽  
Author(s):  
S. S. Dragomir
Keyword(s):  
Hilbert Spaces ◽  
Linear Operators ◽  
Operator Norm ◽  
Inner Product ◽  
Product Spaces ◽  
Numerical Radius ◽  
Inner Product Spaces

In this paper various inequalities between the operator norm and its numerical radius are provided. For this purpose, we employ some classical inequalities for vectors in inner product spaces due to Buzano, Goldstein-Ryff-Clarke, Dragomir-S ´andor and the author.

Topologies on Generalized Inner Product Spaces

1969 ◽  
Vol 21 ◽  
pp. 158-169 ◽  
Author(s):  
Eduard Prugovečki
Keyword(s):  
Hilbert Spaces ◽  
Quantum Physics ◽  
Point Of View ◽  
Inner Product ◽  
Product Spaces ◽  
Topological Vector Spaces ◽  
Fundamental Properties ◽  
Inner Product Spaces ◽  
Algebraic Generalization ◽  
Dirac's Formalism

In the present note we introduce a straightforward algebraic generalization of inner product spaces, which we appropriately name generalized inner product (GIP) spaces. In the same fashion in which different topologies :an be introduced in inner product spaces, adequate topologies can be introduced in GIP spaces in such a manner that topological vector spaces are obtained. We enumerate and derive some fundamental properties of different topologies in GIP spaces, having primarily in mind their possible later application to quantum physics.The desirability of having in quantum physics more general structures than Hilbert spaces (in which quantum mechanics is usually formulated) is suggested by Dirac's formalism (2), which deals with “unnormalizable” vectors. Unfortunately, although this formalism is very elegant from the point of view of the facili ty of dealing with its symbolism, it completely lacks in mathematical rigour.

scholarly journals On Pompeiu Cebysev Type Inequalities for Positive Linear Maps of Selfadjoint Operators in Inner Product Spaces

2018 ◽  
Vol 15 ◽  
pp. 8081-8092
Author(s):  
Mohammad W Alomari
Keyword(s):  
Hilbert Spaces ◽  
Inner Product ◽  
Product Spaces ◽  
Selfadjoint Operators ◽  
Inner Product Spaces ◽  
Linear Maps ◽  
Positive Linear Maps

In this work, generalizations of some inequalities for continuous synchronous (h-asynchronous) functions of linear bounded selfadjoint operators under positive linear maps in Hilbert spaces are proved.

Quasi-inner product spaces of quasi-Sobolev spaces and their completeness

2018 ◽  
pp. 339
Author(s):  
Jawad Kadhim Khalaf Al-Delfi
Keyword(s):  
Hilbert Space ◽  
Sobolev Spaces ◽  
Hilbert Spaces ◽  
Normed Space ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Inner Product Spaces

      Sequences spaces  , m  ,  p  have called quasi-Sobolev spaces were  introduced   by Jawad . K. Al-Delfi in 2013  [1]. In this  paper , we deal with notion of  quasi-inner product  space  by using concept of  quasi-normed  space which is generalized  to normed space and given a  relationship  between  pre-Hilbert space and a  quasi-inner product space with important  results   and   examples.  Completeness properties in quasi-inner   product space gives  us  concept of  quasi-Hilbert space .  We show  that ,  not  all  quasi-Sobolev spaces  ,  are  quasi-Hilbert spaces. The  best  examples which are  quasi-Hilbert spaces and Hilbert spaces  are , where  m  . Finally, propositions, theorems and examples are our own unless otherwise referred.    

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