Riesz Representation Theorem

The Riesz representation theorem is a mathematical theorem that establishes a representation of linear functionals on a Hilbert space. It states that a Hilbert space is its own dual, making the operator theory of Hilbert spaces more manageable than other infinite dimensional Banach spaces. The theorem's proof is based on a lemma that shows how to represent a linear functional by demonstrating certain conditions.From: Functional Analysis and Summability [2020], A First Course in Functional Analysis [2017], Modern Analysis [2017]

Hilbert Spaces

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Published in Eberhard Malkowsky, Vladimir Rakočević, Advanced Functional Analysis, 2019

Eberhard Malkowsky, Vladimir Rakočević

(Riesz representation theorem)If f is a continuous linear functional on a Hilbert space X then there exists a unique vectory∈Xsuch thatf(x)=〈x,y〉forallx∈X.

The inverse scattering problem for a conductive boundary condition and transmission eigenvalues

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Published in Applicable Analysis, 2020

I. Harris, A. Kleefeld

The transmission eigenvalue problem can now be written in operator form as: find the values such that there is a nontrivial solution satisfying where the bounded linear operators and are defined by the Riesz representation theorem such that for all . We see that the operator is compact by appealing to Rellich's embedding theorem and the compact embedding of into .

Existence, uniqueness and regularity of piezoelectric partial differential equations

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Published in Applicable Analysis, 2022

Benjamin Jurgelucks, Veronika Schulze, Tom Lahmer

Weak solutions are functions and as in Equations (2) and (3) where such that for almost all for all the following equation holds: with and Note that by the Riesz representation theorem there exists a unique representation for the latter functionals as an inner product, i.e. and . As is common in the field of partial differential equation for convenience we will also use the same symbols f and g to refer to the Riesz-representative as well as the functionals and . Furthermore, we remember that and that , e are constant. The integrals of the right-hand side , are finite, their values depend, e.g. only on domain Ω but not on time t. Thus, by integrating this constant value over time we can estimate the Bochner-space norm of f by and analogously we get

Riesz Representation Theorem

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Hilbert Spaces

The inverse scattering problem for a conductive boundary condition and transmission eigenvalues

Existence, uniqueness and regularity of piezoelectric partial differential equations