Achievability under arbitrary norm

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Assume we have some function that has an achievable upper bound given by a linear norm in a finite dimensional vector space. Can we state for sure that the bound is also achievable for the same vector space but equipped with arbitrary Lp-norm (Hölder's norm)?



Thank you for any hint!




functional-analysis normed-spaces




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  • 1




    I'm not sure I understand what you mean here. For starters, what is the domain and codomain of your function?
    – Lorenzo Quarisa
    Aug 26 at 8:51










  • in the simplest case, I assume a single objective real-valued function F(C, IICII), with known analytical expression.. Here C is n-dimensional (n is finite) real valued vector in a normed vector space, where IICII is a p-norm (i.e. the Hölder norm). And we are able to prove that F(C, IICII) is less than or equal to IICII for any C. So IICII is an upper bound.. We can also show that there exists a particular C* such that F(C*, IICII) equals to IICII, if IIC*II is either Chebyshev (Lmax) or linear norm (L1)..
    – Yury Nikulin
    Aug 27 at 6:50










  • The question is the following: if there is any theretical result allowing us to claim that IICII is a attainable upper bound no matter what p-norm is used, i.e. the upper bound is atainable for all p, and not only for p=1 (linear) or p=infinity (Chebushev)
    – Yury Nikulin
    Aug 27 at 6:50














up vote
0
down vote

favorite












Assume we have some function that has an achievable upper bound given by a linear norm in a finite dimensional vector space. Can we state for sure that the bound is also achievable for the same vector space but equipped with arbitrary Lp-norm (Hölder's norm)?



Thank you for any hint!




functional-analysis normed-spaces




share|cite|improve this question
















  • 1




    I'm not sure I understand what you mean here. For starters, what is the domain and codomain of your function?
    – Lorenzo Quarisa
    Aug 26 at 8:51










  • in the simplest case, I assume a single objective real-valued function F(C, IICII), with known analytical expression.. Here C is n-dimensional (n is finite) real valued vector in a normed vector space, where IICII is a p-norm (i.e. the Hölder norm). And we are able to prove that F(C, IICII) is less than or equal to IICII for any C. So IICII is an upper bound.. We can also show that there exists a particular C* such that F(C*, IICII) equals to IICII, if IIC*II is either Chebyshev (Lmax) or linear norm (L1)..
    – Yury Nikulin
    Aug 27 at 6:50










  • The question is the following: if there is any theretical result allowing us to claim that IICII is a attainable upper bound no matter what p-norm is used, i.e. the upper bound is atainable for all p, and not only for p=1 (linear) or p=infinity (Chebushev)
    – Yury Nikulin
    Aug 27 at 6:50












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Assume we have some function that has an achievable upper bound given by a linear norm in a finite dimensional vector space. Can we state for sure that the bound is also achievable for the same vector space but equipped with arbitrary Lp-norm (Hölder's norm)?



Thank you for any hint!




functional-analysis normed-spaces




share|cite|improve this question












Assume we have some function that has an achievable upper bound given by a linear norm in a finite dimensional vector space. Can we state for sure that the bound is also achievable for the same vector space but equipped with arbitrary Lp-norm (Hölder's norm)?



Thank you for any hint!





functional-analysis normed-spaces





share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Aug 26 at 8:27









Yury Nikulin

1




1







  • 1




    I'm not sure I understand what you mean here. For starters, what is the domain and codomain of your function?
    – Lorenzo Quarisa
    Aug 26 at 8:51










  • in the simplest case, I assume a single objective real-valued function F(C, IICII), with known analytical expression.. Here C is n-dimensional (n is finite) real valued vector in a normed vector space, where IICII is a p-norm (i.e. the Hölder norm). And we are able to prove that F(C, IICII) is less than or equal to IICII for any C. So IICII is an upper bound.. We can also show that there exists a particular C* such that F(C*, IICII) equals to IICII, if IIC*II is either Chebyshev (Lmax) or linear norm (L1)..
    – Yury Nikulin
    Aug 27 at 6:50










  • The question is the following: if there is any theretical result allowing us to claim that IICII is a attainable upper bound no matter what p-norm is used, i.e. the upper bound is atainable for all p, and not only for p=1 (linear) or p=infinity (Chebushev)
    – Yury Nikulin
    Aug 27 at 6:50












  • 1




    I'm not sure I understand what you mean here. For starters, what is the domain and codomain of your function?
    – Lorenzo Quarisa
    Aug 26 at 8:51










  • in the simplest case, I assume a single objective real-valued function F(C, IICII), with known analytical expression.. Here C is n-dimensional (n is finite) real valued vector in a normed vector space, where IICII is a p-norm (i.e. the Hölder norm). And we are able to prove that F(C, IICII) is less than or equal to IICII for any C. So IICII is an upper bound.. We can also show that there exists a particular C* such that F(C*, IICII) equals to IICII, if IIC*II is either Chebyshev (Lmax) or linear norm (L1)..
    – Yury Nikulin
    Aug 27 at 6:50










  • The question is the following: if there is any theretical result allowing us to claim that IICII is a attainable upper bound no matter what p-norm is used, i.e. the upper bound is atainable for all p, and not only for p=1 (linear) or p=infinity (Chebushev)
    – Yury Nikulin
    Aug 27 at 6:50







1




1




I'm not sure I understand what you mean here. For starters, what is the domain and codomain of your function?
– Lorenzo Quarisa
Aug 26 at 8:51




I'm not sure I understand what you mean here. For starters, what is the domain and codomain of your function?
– Lorenzo Quarisa
Aug 26 at 8:51












in the simplest case, I assume a single objective real-valued function F(C, IICII), with known analytical expression.. Here C is n-dimensional (n is finite) real valued vector in a normed vector space, where IICII is a p-norm (i.e. the Hölder norm). And we are able to prove that F(C, IICII) is less than or equal to IICII for any C. So IICII is an upper bound.. We can also show that there exists a particular C* such that F(C*, IICII) equals to IICII, if IIC*II is either Chebyshev (Lmax) or linear norm (L1)..
– Yury Nikulin
Aug 27 at 6:50




in the simplest case, I assume a single objective real-valued function F(C, IICII), with known analytical expression.. Here C is n-dimensional (n is finite) real valued vector in a normed vector space, where IICII is a p-norm (i.e. the Hölder norm). And we are able to prove that F(C, IICII) is less than or equal to IICII for any C. So IICII is an upper bound.. We can also show that there exists a particular C* such that F(C*, IICII) equals to IICII, if IIC*II is either Chebyshev (Lmax) or linear norm (L1)..
– Yury Nikulin
Aug 27 at 6:50












The question is the following: if there is any theretical result allowing us to claim that IICII is a attainable upper bound no matter what p-norm is used, i.e. the upper bound is atainable for all p, and not only for p=1 (linear) or p=infinity (Chebushev)
– Yury Nikulin
Aug 27 at 6:50




The question is the following: if there is any theretical result allowing us to claim that IICII is a attainable upper bound no matter what p-norm is used, i.e. the upper bound is atainable for all p, and not only for p=1 (linear) or p=infinity (Chebushev)
– Yury Nikulin
Aug 27 at 6:50















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