Bounded linear functional separating a hyperplane and a point












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I want to find examples of normed linear space, a convex subset $C$ of $X$ and a point $x_0in Xsetminus C$ such that for all bounded linear functionals $f:Xto mathbb{K}$, Re$f(x_0)<sup{text{Re}f(x):xin C}$.



If I start with a proper dense subspace $C$ of $X$, then there exists a sequence $(x_n)$ in $C$ such that $x_nto x_0$. Thus for all bounded linear functional $f$ on $X$, Re$f(x_n)to text{Re}f(x_0)$. Thus Re$f(x_0)leq sup{text{Re}f(x):xin C}$. But how to show the strict inequality?






functional-analysis convex-analysis







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    3












    $begingroup$


    I want to find examples of normed linear space, a convex subset $C$ of $X$ and a point $x_0in Xsetminus C$ such that for all bounded linear functionals $f:Xto mathbb{K}$, Re$f(x_0)<sup{text{Re}f(x):xin C}$.



    If I start with a proper dense subspace $C$ of $X$, then there exists a sequence $(x_n)$ in $C$ such that $x_nto x_0$. Thus for all bounded linear functional $f$ on $X$, Re$f(x_n)to text{Re}f(x_0)$. Thus Re$f(x_0)leq sup{text{Re}f(x):xin C}$. But how to show the strict inequality?






    functional-analysis convex-analysis







    share|cite|improve this question









    $endgroup$















      3












      3








      3





      $begingroup$


      I want to find examples of normed linear space, a convex subset $C$ of $X$ and a point $x_0in Xsetminus C$ such that for all bounded linear functionals $f:Xto mathbb{K}$, Re$f(x_0)<sup{text{Re}f(x):xin C}$.



      If I start with a proper dense subspace $C$ of $X$, then there exists a sequence $(x_n)$ in $C$ such that $x_nto x_0$. Thus for all bounded linear functional $f$ on $X$, Re$f(x_n)to text{Re}f(x_0)$. Thus Re$f(x_0)leq sup{text{Re}f(x):xin C}$. But how to show the strict inequality?






      functional-analysis convex-analysis







      share|cite|improve this question









      $endgroup$




      I want to find examples of normed linear space, a convex subset $C$ of $X$ and a point $x_0in Xsetminus C$ such that for all bounded linear functionals $f:Xto mathbb{K}$, Re$f(x_0)<sup{text{Re}f(x):xin C}$.



      If I start with a proper dense subspace $C$ of $X$, then there exists a sequence $(x_n)$ in $C$ such that $x_nto x_0$. Thus for all bounded linear functional $f$ on $X$, Re$f(x_n)to text{Re}f(x_0)$. Thus Re$f(x_0)leq sup{text{Re}f(x):xin C}$. But how to show the strict inequality?






      functional-analysis convex-analysis




      functional-analysis convex-analysis






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Feb 3 at 3:38









      AnupamAnupam

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