General form of linear continuous functional and criterion of the best approximant in spaces with mixed integral metric

2014 ◽  
Vol 22 ◽  
pp. 91
Author(s):  
V.M. Traktyns'ka ◽  
M.Ye. Tkachenko
Keyword(s):  
Linear Functional ◽  
Continuous Functional ◽  
Bounded Linear ◽  
Linear Continuous Functional ◽  
Bounded Linear Functional

The questions of the characterization of the best approximant in spaces with mixed integral metric with weight were considered in this article. The general form of a bounded linear functional and the criterion of best approximant in these spaces are obtained.

scholarly journals The characterization of the best approximant for the multivariable functions in the space $L_{p_1,..,p_n;\Omega }$

2020 ◽  
Vol 27 (2) ◽  
pp. 36
Author(s):  
M.Ye. Tkachenko ◽  
V.M. Traktynska
Keyword(s):  
Linear Functional ◽  
Bounded Linear ◽  
Bounded Linear Functional

The questions of the characterization of the best approximant in spaces of multivariable functions with mixed integral metric with weight were considered in this article. The general form of a bounded linear functional and the criterion of the best approximant in these spaces are obtained.

scholarly journals Which solutions of the third problem for the Poisson equation are bounded?

2004 ◽  
Vol 2004 (6) ◽  
pp. 501-510
Author(s):  
Dagmar Medková
Keyword(s):  
Boundary Condition ◽  
Weak Solution ◽  
Bounded Domain ◽  
Linear Functional ◽  
Nonnegative Function ◽  
Newtonian Potential ◽  
Bounded Linear ◽  
The Third ◽  
The Poisson Equation ◽  
Bounded Linear Functional

This paper deals with the problemΔu=gonGand∂u/∂n+uf=Lon∂G. Here,G⊂ℝm,m>2, is a bounded domain with Lyapunov boundary,fis a bounded nonnegative function on the boundary ofG,Lis a bounded linear functional onW1,2(G)representable by a real measureμon the boundary ofG, andg∈L2(G)∩Lp(G),p>m/2. It is shown that a weak solution of this problem is bounded inGif and only if the Newtonian potential corresponding to the boundary conditionμis bounded inG.

Uniqueness of norm-preserving extensions of functionals on the space of compact operators

2019 ◽  
Vol 125 (1) ◽  
pp. 67-83
Author(s):  
Julia Martsinkevitš ◽  
Märt Põldvere
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Linear Functional ◽  
Dual Space ◽  
Closed Subspace ◽  
Compact Operators ◽  
Bounded Linear ◽  
Nikodym Property ◽  
Space Of Compact Operators ◽  
Bounded Linear Functional

Godefroy, Kalton, and Saphar called a closed subspace $Y$ of a Banach space $Z$ an ideal if its annihilator $Y^\bot $ is the kernel of a norm-one projection $P$ on the dual space $Z^\ast $. If $Y$ is an ideal in $Z$ with respect to a projection on $Z^\ast $ whose range is norming for $Z$, then $Y$ is said to be a strict ideal. We study uniqueness of norm-preserving extensions of functionals on the space $\mathcal{K}(X,Y) $ of compact operators between Banach spaces $X$ and $Y$ to the larger space $\mathcal{K}(X,Z) $ under the assumption that $Y$ is a strict ideal in $Z$. Our main results are: (1) if $y^\ast $ is an extreme point of $B_{Y^{\ast} }$ having a unique norm-preserving extension to $Z$, and $x^{\ast\ast} \in B_{X^{\ast\ast} }$, then the only norm-preserving extension of the functional $x^{\ast\ast} \otimes y^\ast \in \mathcal {K}(X,Y)^\ast $ to $\mathcal {K}(X,Z)$ is $x^{\ast\ast} \otimes z^\ast $ where $z^\ast \in Z^\ast $ is the only norm-preserving extension of $y^\ast $ to $Z$; (2) if $\mathcal{K}(X,Y) $ is an ideal in $\mathcal{K}(X,Z) $ and $Y$ has Phelps' property $U$ in its bidual $Y^{\ast\ast} $ (i.e., every bounded linear functional on $Y$ admits a unique norm-preserving extension to $Y^{\ast\ast} $), then $\mathcal{K}(X,Y) $ has property $U$ in $\mathcal{K}(X,Z) $ whenever $X^{\ast\ast} $ has the Radon-Nikodým property.

scholarly journals A note on the Følner condition for amenability

1993 ◽  
Vol 131 ◽  
pp. 67-74 ◽  
Author(s):  
Toshiaki Adachi
Keyword(s):  
Solvable Group ◽  
Finite Groups ◽  
Linear Functional ◽  
Discrete Group ◽  
The Other ◽  
Invariant Mean ◽  
Bounded Linear ◽  
Subexponential Growth ◽  
Bounded Functions ◽  
Bounded Linear Functional

Let G be a countably generated discrete group. A right-invariant mean μ on G is a bounded linear functional of the space L∞(G) of bounded functions on G having the property:We say that G is amenable if it is equipped with a right-invariant mean. Finite groups, abelian groups, in fact, groups of subexponential growth are amenable. Solvable group are also amenable. Subgroups and quotients of amenable groups are amenable. On the other hand, free groups having two generators and over are non-amenable.

scholarly journals n-th Order Functional Problems with Resonance of Dimension One

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2384
Author(s):  
Erin Benham ◽  
Nickolai Kosmatov
Keyword(s):  
Boundary Value Problem ◽  
Linear Functional ◽  
Boundary Value ◽  
Coincidence Degree ◽  
Constructive Method ◽  
Bounded Linear ◽  
Dimension One ◽  
Bounded Linear Functional ◽  
Existence Criteria ◽  
Functional Problems

We consider the nonlinear n-th order boundary value problem Lu=u(n)=f(t,u(t),u′(t),…,u(n−1)(t))=Nu given arbitrary bounded linear functional conditions Bi(u)=0, i=1,…,n and develop a method that allows us to study all such resonance problems of order one, as well as implementing a more general constructive method for deriving existence criteria in the framework of the coincidence degree method of Mawhin. We demonstrate applicability of the formalism by giving an example for n=4.

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