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Math Help - norms on R

  1. #1
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    norms on R

    Describe all norms on \mathbb{R}^1. Does anyone know how to solve it?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: norms on R

    Quote Originally Posted by Camille91 View Post
    Describe all norms on \mathbb{R}^1. Does anyone know how to solve it?
    Think about it like this, let f:\mathbb{R}\to\mathbb{R} be a norm, then we know that f(ar)=|a|f(r) for every a\in\mathbb{R}, and so evidently f(r)=|r|f(1) for all r\in\mathbb{R}. Thus, if f_\lambda:\mathbb{R}\to\mathbb{R} denotes the function f_\lambda(x)=|x|\lambda then evidently every norm is a f_\lambda. Now, \lambda|x+y|\leqslant \lambda|x|+\lambda|y| holds for all x,y\in\mathbb{R} if and only if \lambda\geqslant 0. Noting finally that f_\lambda(0)=0 we may combine this to conclude that \left\{\text{norms}\right\}=\left\{f_\lambda :\lambda\geqslant 0\right\}.
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    Re: norms on R

    I think you are right. Just using simple definiton of norms. Thanks a lot!
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