An “open-strict” Version of Hahn Banach Separation Theorem?












2














Is the following statement true?



Let $X$ be a real linear space, $A,B subset X$ two disjoint convex sets with the following "algebraic openness" property: Every $x in A$ is an internal point of A, and so is every point of B and internal point of B. Then there exists a linear functional $f:X rightarrow mathbb{R}$ and $t in mathbb{R}$ such that for all $x in A, y in B$ we have
$$ f(x) < t < f(y)$$



A similar version (but in the context of TVS and one of the sets is open) is given in Wikipedia, Note that it only admits a "half strict" separation. I believe that it is still true when topological openness is replaced by the "algebraic openness" definition given here, but the question is if I assume both sets are "algebraically open", is it true that I can get a strict separation from both sides? Any help is appreciated.










share|cite|improve this question



























    2














    Is the following statement true?



    Let $X$ be a real linear space, $A,B subset X$ two disjoint convex sets with the following "algebraic openness" property: Every $x in A$ is an internal point of A, and so is every point of B and internal point of B. Then there exists a linear functional $f:X rightarrow mathbb{R}$ and $t in mathbb{R}$ such that for all $x in A, y in B$ we have
    $$ f(x) < t < f(y)$$



    A similar version (but in the context of TVS and one of the sets is open) is given in Wikipedia, Note that it only admits a "half strict" separation. I believe that it is still true when topological openness is replaced by the "algebraic openness" definition given here, but the question is if I assume both sets are "algebraically open", is it true that I can get a strict separation from both sides? Any help is appreciated.










    share|cite|improve this question

























      2












      2








      2


      1





      Is the following statement true?



      Let $X$ be a real linear space, $A,B subset X$ two disjoint convex sets with the following "algebraic openness" property: Every $x in A$ is an internal point of A, and so is every point of B and internal point of B. Then there exists a linear functional $f:X rightarrow mathbb{R}$ and $t in mathbb{R}$ such that for all $x in A, y in B$ we have
      $$ f(x) < t < f(y)$$



      A similar version (but in the context of TVS and one of the sets is open) is given in Wikipedia, Note that it only admits a "half strict" separation. I believe that it is still true when topological openness is replaced by the "algebraic openness" definition given here, but the question is if I assume both sets are "algebraically open", is it true that I can get a strict separation from both sides? Any help is appreciated.










      share|cite|improve this question













      Is the following statement true?



      Let $X$ be a real linear space, $A,B subset X$ two disjoint convex sets with the following "algebraic openness" property: Every $x in A$ is an internal point of A, and so is every point of B and internal point of B. Then there exists a linear functional $f:X rightarrow mathbb{R}$ and $t in mathbb{R}$ such that for all $x in A, y in B$ we have
      $$ f(x) < t < f(y)$$



      A similar version (but in the context of TVS and one of the sets is open) is given in Wikipedia, Note that it only admits a "half strict" separation. I believe that it is still true when topological openness is replaced by the "algebraic openness" definition given here, but the question is if I assume both sets are "algebraically open", is it true that I can get a strict separation from both sides? Any help is appreciated.







      functional-analysis convex-analysis hahn-banach-theorem






      share|cite|improve this question













      share|cite|improve this question











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      share|cite|improve this question










      asked Jan 1 at 20:21









      pitariver

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          IMPORTANT EDIT: A theorem related to the question stated in the OP can be found here. Theorem 4 states: If $A$ and $B$ are disjoint convex sets in $X$ and $A$ has an internal point, then $A$ and $B$ can be (weakly) separated. That is, there exists $f:Xrightarrowmathbb{R}$ such that $$sup_{ain A}f(a)leqinf_{bin B}f(b).$$



          EDIT 2: We can use this theorem to answer the question in the OP. This follows from the following proposition.



          Proposition: Let $f:Xtomathbb{R}$ be linear and non-zero. Then for any $A$ obeying the "algebraic openness" property, we have that $f(A)$ is open.



          Proof: Let $tin f(A)$. So $f(a)=t$ for some $ain A$. Because $f$ is non-zero, we find some $xin X$ such that $f(x)>0$. By the "algebraic openness" property of $A$, there exists $varepsilon>0$ such that $a+(-varepsilon,varepsilon)cdot xsubset A$. Hence, $(t-varepsilon f(x),t+varepsilon f(x))subseteq f(A)$, so $f(A)$ is open.



          Combining the two results, we have for all $ain A$ and $bin B$ that $$f(a)<sup_{alphain A}f(alpha)leqinf_{betain B}f(beta)<f(b).$$



          ORIGINAL ANSWER: The set of all sets with your "algebraic openness" property makes $X$ a topological vector space. Hence, if $A$ and $B$ are disjoint, convex and open in this topology, then there exists a continuous linear functional $phi$ and a constant $sinmathbb{R}$ such that $phi(a)<sleqphi(b)$ for all $ain A$ and $bin B$. But there also exists a continuous linear functional $psi$ and a constant $tinmathbb{R}$ such that $psi(b)<tleqpsi(a)$ for all $bin B$ and $ain A$. Then $f:=phi-psi$ is a continuous linear functional such that $f(a)<s-t<f(b)$ holds for all $ain A$ and $bin B$.



          EDIT 3: We now know that the "algebraic openness" property does not define a topological vector space. Do algebraically open sets define a vector space topology?






          share|cite|improve this answer























          • Roughly speaking, scalar multiplication is continuous exactly by the "algebraically openness" property, and vector addition is continuous because the property is invariant under translation.
            – SmileyCraft
            2 days ago










          • @SimleyCraft I a m still not sure it's necessarily true (in anycase I understand how to solve the problem with the $phi - psi$ argument). For example why should scalar mult be continues? let $x_0 in X, lambda_0 in mathbb{R}, lambda_0 x_0 in U$ open, you would want a neighborhood of $x_0$, V, for which there exists $delta>0$, $vert lambda - lambda_0 vert < delta implies lambda V subset U$
            – pitariver
            2 days ago












          • @pitariver I must admit I underestimated the difficulty of proving this fact. I did already realize the problems you mentioned, but I had some ideas of proving this, however they did not work after all. I edited the post with a new answer. However I would still love to know whether the "algebraic openness" defines a linear topology.
            – SmileyCraft
            2 days ago










          • Thanks for the link, though sadly, it still doesn't solve the problem: I asked about a strict separation, yet Theorem 4 in the paper only guarantees "weak" separation(which I already heard of). The counter example given immediately after the theorem also doesn't tell us much about this case. I am starting to wonder whether the statement is actually true, even for the "half strict" case.
            – pitariver
            yesterday












          • Also in regards to the "algebraically open" sets defining a TVS, I am not entirely sure it's true. Thanks for the attention so far!
            – pitariver
            yesterday











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          IMPORTANT EDIT: A theorem related to the question stated in the OP can be found here. Theorem 4 states: If $A$ and $B$ are disjoint convex sets in $X$ and $A$ has an internal point, then $A$ and $B$ can be (weakly) separated. That is, there exists $f:Xrightarrowmathbb{R}$ such that $$sup_{ain A}f(a)leqinf_{bin B}f(b).$$



          EDIT 2: We can use this theorem to answer the question in the OP. This follows from the following proposition.



          Proposition: Let $f:Xtomathbb{R}$ be linear and non-zero. Then for any $A$ obeying the "algebraic openness" property, we have that $f(A)$ is open.



          Proof: Let $tin f(A)$. So $f(a)=t$ for some $ain A$. Because $f$ is non-zero, we find some $xin X$ such that $f(x)>0$. By the "algebraic openness" property of $A$, there exists $varepsilon>0$ such that $a+(-varepsilon,varepsilon)cdot xsubset A$. Hence, $(t-varepsilon f(x),t+varepsilon f(x))subseteq f(A)$, so $f(A)$ is open.



          Combining the two results, we have for all $ain A$ and $bin B$ that $$f(a)<sup_{alphain A}f(alpha)leqinf_{betain B}f(beta)<f(b).$$



          ORIGINAL ANSWER: The set of all sets with your "algebraic openness" property makes $X$ a topological vector space. Hence, if $A$ and $B$ are disjoint, convex and open in this topology, then there exists a continuous linear functional $phi$ and a constant $sinmathbb{R}$ such that $phi(a)<sleqphi(b)$ for all $ain A$ and $bin B$. But there also exists a continuous linear functional $psi$ and a constant $tinmathbb{R}$ such that $psi(b)<tleqpsi(a)$ for all $bin B$ and $ain A$. Then $f:=phi-psi$ is a continuous linear functional such that $f(a)<s-t<f(b)$ holds for all $ain A$ and $bin B$.



          EDIT 3: We now know that the "algebraic openness" property does not define a topological vector space. Do algebraically open sets define a vector space topology?






          share|cite|improve this answer























          • Roughly speaking, scalar multiplication is continuous exactly by the "algebraically openness" property, and vector addition is continuous because the property is invariant under translation.
            – SmileyCraft
            2 days ago










          • @SimleyCraft I a m still not sure it's necessarily true (in anycase I understand how to solve the problem with the $phi - psi$ argument). For example why should scalar mult be continues? let $x_0 in X, lambda_0 in mathbb{R}, lambda_0 x_0 in U$ open, you would want a neighborhood of $x_0$, V, for which there exists $delta>0$, $vert lambda - lambda_0 vert < delta implies lambda V subset U$
            – pitariver
            2 days ago












          • @pitariver I must admit I underestimated the difficulty of proving this fact. I did already realize the problems you mentioned, but I had some ideas of proving this, however they did not work after all. I edited the post with a new answer. However I would still love to know whether the "algebraic openness" defines a linear topology.
            – SmileyCraft
            2 days ago










          • Thanks for the link, though sadly, it still doesn't solve the problem: I asked about a strict separation, yet Theorem 4 in the paper only guarantees "weak" separation(which I already heard of). The counter example given immediately after the theorem also doesn't tell us much about this case. I am starting to wonder whether the statement is actually true, even for the "half strict" case.
            – pitariver
            yesterday












          • Also in regards to the "algebraically open" sets defining a TVS, I am not entirely sure it's true. Thanks for the attention so far!
            – pitariver
            yesterday
















          1














          IMPORTANT EDIT: A theorem related to the question stated in the OP can be found here. Theorem 4 states: If $A$ and $B$ are disjoint convex sets in $X$ and $A$ has an internal point, then $A$ and $B$ can be (weakly) separated. That is, there exists $f:Xrightarrowmathbb{R}$ such that $$sup_{ain A}f(a)leqinf_{bin B}f(b).$$



          EDIT 2: We can use this theorem to answer the question in the OP. This follows from the following proposition.



          Proposition: Let $f:Xtomathbb{R}$ be linear and non-zero. Then for any $A$ obeying the "algebraic openness" property, we have that $f(A)$ is open.



          Proof: Let $tin f(A)$. So $f(a)=t$ for some $ain A$. Because $f$ is non-zero, we find some $xin X$ such that $f(x)>0$. By the "algebraic openness" property of $A$, there exists $varepsilon>0$ such that $a+(-varepsilon,varepsilon)cdot xsubset A$. Hence, $(t-varepsilon f(x),t+varepsilon f(x))subseteq f(A)$, so $f(A)$ is open.



          Combining the two results, we have for all $ain A$ and $bin B$ that $$f(a)<sup_{alphain A}f(alpha)leqinf_{betain B}f(beta)<f(b).$$



          ORIGINAL ANSWER: The set of all sets with your "algebraic openness" property makes $X$ a topological vector space. Hence, if $A$ and $B$ are disjoint, convex and open in this topology, then there exists a continuous linear functional $phi$ and a constant $sinmathbb{R}$ such that $phi(a)<sleqphi(b)$ for all $ain A$ and $bin B$. But there also exists a continuous linear functional $psi$ and a constant $tinmathbb{R}$ such that $psi(b)<tleqpsi(a)$ for all $bin B$ and $ain A$. Then $f:=phi-psi$ is a continuous linear functional such that $f(a)<s-t<f(b)$ holds for all $ain A$ and $bin B$.



          EDIT 3: We now know that the "algebraic openness" property does not define a topological vector space. Do algebraically open sets define a vector space topology?






          share|cite|improve this answer























          • Roughly speaking, scalar multiplication is continuous exactly by the "algebraically openness" property, and vector addition is continuous because the property is invariant under translation.
            – SmileyCraft
            2 days ago










          • @SimleyCraft I a m still not sure it's necessarily true (in anycase I understand how to solve the problem with the $phi - psi$ argument). For example why should scalar mult be continues? let $x_0 in X, lambda_0 in mathbb{R}, lambda_0 x_0 in U$ open, you would want a neighborhood of $x_0$, V, for which there exists $delta>0$, $vert lambda - lambda_0 vert < delta implies lambda V subset U$
            – pitariver
            2 days ago












          • @pitariver I must admit I underestimated the difficulty of proving this fact. I did already realize the problems you mentioned, but I had some ideas of proving this, however they did not work after all. I edited the post with a new answer. However I would still love to know whether the "algebraic openness" defines a linear topology.
            – SmileyCraft
            2 days ago










          • Thanks for the link, though sadly, it still doesn't solve the problem: I asked about a strict separation, yet Theorem 4 in the paper only guarantees "weak" separation(which I already heard of). The counter example given immediately after the theorem also doesn't tell us much about this case. I am starting to wonder whether the statement is actually true, even for the "half strict" case.
            – pitariver
            yesterday












          • Also in regards to the "algebraically open" sets defining a TVS, I am not entirely sure it's true. Thanks for the attention so far!
            – pitariver
            yesterday














          1












          1








          1






          IMPORTANT EDIT: A theorem related to the question stated in the OP can be found here. Theorem 4 states: If $A$ and $B$ are disjoint convex sets in $X$ and $A$ has an internal point, then $A$ and $B$ can be (weakly) separated. That is, there exists $f:Xrightarrowmathbb{R}$ such that $$sup_{ain A}f(a)leqinf_{bin B}f(b).$$



          EDIT 2: We can use this theorem to answer the question in the OP. This follows from the following proposition.



          Proposition: Let $f:Xtomathbb{R}$ be linear and non-zero. Then for any $A$ obeying the "algebraic openness" property, we have that $f(A)$ is open.



          Proof: Let $tin f(A)$. So $f(a)=t$ for some $ain A$. Because $f$ is non-zero, we find some $xin X$ such that $f(x)>0$. By the "algebraic openness" property of $A$, there exists $varepsilon>0$ such that $a+(-varepsilon,varepsilon)cdot xsubset A$. Hence, $(t-varepsilon f(x),t+varepsilon f(x))subseteq f(A)$, so $f(A)$ is open.



          Combining the two results, we have for all $ain A$ and $bin B$ that $$f(a)<sup_{alphain A}f(alpha)leqinf_{betain B}f(beta)<f(b).$$



          ORIGINAL ANSWER: The set of all sets with your "algebraic openness" property makes $X$ a topological vector space. Hence, if $A$ and $B$ are disjoint, convex and open in this topology, then there exists a continuous linear functional $phi$ and a constant $sinmathbb{R}$ such that $phi(a)<sleqphi(b)$ for all $ain A$ and $bin B$. But there also exists a continuous linear functional $psi$ and a constant $tinmathbb{R}$ such that $psi(b)<tleqpsi(a)$ for all $bin B$ and $ain A$. Then $f:=phi-psi$ is a continuous linear functional such that $f(a)<s-t<f(b)$ holds for all $ain A$ and $bin B$.



          EDIT 3: We now know that the "algebraic openness" property does not define a topological vector space. Do algebraically open sets define a vector space topology?






          share|cite|improve this answer














          IMPORTANT EDIT: A theorem related to the question stated in the OP can be found here. Theorem 4 states: If $A$ and $B$ are disjoint convex sets in $X$ and $A$ has an internal point, then $A$ and $B$ can be (weakly) separated. That is, there exists $f:Xrightarrowmathbb{R}$ such that $$sup_{ain A}f(a)leqinf_{bin B}f(b).$$



          EDIT 2: We can use this theorem to answer the question in the OP. This follows from the following proposition.



          Proposition: Let $f:Xtomathbb{R}$ be linear and non-zero. Then for any $A$ obeying the "algebraic openness" property, we have that $f(A)$ is open.



          Proof: Let $tin f(A)$. So $f(a)=t$ for some $ain A$. Because $f$ is non-zero, we find some $xin X$ such that $f(x)>0$. By the "algebraic openness" property of $A$, there exists $varepsilon>0$ such that $a+(-varepsilon,varepsilon)cdot xsubset A$. Hence, $(t-varepsilon f(x),t+varepsilon f(x))subseteq f(A)$, so $f(A)$ is open.



          Combining the two results, we have for all $ain A$ and $bin B$ that $$f(a)<sup_{alphain A}f(alpha)leqinf_{betain B}f(beta)<f(b).$$



          ORIGINAL ANSWER: The set of all sets with your "algebraic openness" property makes $X$ a topological vector space. Hence, if $A$ and $B$ are disjoint, convex and open in this topology, then there exists a continuous linear functional $phi$ and a constant $sinmathbb{R}$ such that $phi(a)<sleqphi(b)$ for all $ain A$ and $bin B$. But there also exists a continuous linear functional $psi$ and a constant $tinmathbb{R}$ such that $psi(b)<tleqpsi(a)$ for all $bin B$ and $ain A$. Then $f:=phi-psi$ is a continuous linear functional such that $f(a)<s-t<f(b)$ holds for all $ain A$ and $bin B$.



          EDIT 3: We now know that the "algebraic openness" property does not define a topological vector space. Do algebraically open sets define a vector space topology?







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited yesterday

























          answered Jan 1 at 23:13









          SmileyCraft

          2,971416




          2,971416












          • Roughly speaking, scalar multiplication is continuous exactly by the "algebraically openness" property, and vector addition is continuous because the property is invariant under translation.
            – SmileyCraft
            2 days ago










          • @SimleyCraft I a m still not sure it's necessarily true (in anycase I understand how to solve the problem with the $phi - psi$ argument). For example why should scalar mult be continues? let $x_0 in X, lambda_0 in mathbb{R}, lambda_0 x_0 in U$ open, you would want a neighborhood of $x_0$, V, for which there exists $delta>0$, $vert lambda - lambda_0 vert < delta implies lambda V subset U$
            – pitariver
            2 days ago












          • @pitariver I must admit I underestimated the difficulty of proving this fact. I did already realize the problems you mentioned, but I had some ideas of proving this, however they did not work after all. I edited the post with a new answer. However I would still love to know whether the "algebraic openness" defines a linear topology.
            – SmileyCraft
            2 days ago










          • Thanks for the link, though sadly, it still doesn't solve the problem: I asked about a strict separation, yet Theorem 4 in the paper only guarantees "weak" separation(which I already heard of). The counter example given immediately after the theorem also doesn't tell us much about this case. I am starting to wonder whether the statement is actually true, even for the "half strict" case.
            – pitariver
            yesterday












          • Also in regards to the "algebraically open" sets defining a TVS, I am not entirely sure it's true. Thanks for the attention so far!
            – pitariver
            yesterday


















          • Roughly speaking, scalar multiplication is continuous exactly by the "algebraically openness" property, and vector addition is continuous because the property is invariant under translation.
            – SmileyCraft
            2 days ago










          • @SimleyCraft I a m still not sure it's necessarily true (in anycase I understand how to solve the problem with the $phi - psi$ argument). For example why should scalar mult be continues? let $x_0 in X, lambda_0 in mathbb{R}, lambda_0 x_0 in U$ open, you would want a neighborhood of $x_0$, V, for which there exists $delta>0$, $vert lambda - lambda_0 vert < delta implies lambda V subset U$
            – pitariver
            2 days ago












          • @pitariver I must admit I underestimated the difficulty of proving this fact. I did already realize the problems you mentioned, but I had some ideas of proving this, however they did not work after all. I edited the post with a new answer. However I would still love to know whether the "algebraic openness" defines a linear topology.
            – SmileyCraft
            2 days ago










          • Thanks for the link, though sadly, it still doesn't solve the problem: I asked about a strict separation, yet Theorem 4 in the paper only guarantees "weak" separation(which I already heard of). The counter example given immediately after the theorem also doesn't tell us much about this case. I am starting to wonder whether the statement is actually true, even for the "half strict" case.
            – pitariver
            yesterday












          • Also in regards to the "algebraically open" sets defining a TVS, I am not entirely sure it's true. Thanks for the attention so far!
            – pitariver
            yesterday
















          Roughly speaking, scalar multiplication is continuous exactly by the "algebraically openness" property, and vector addition is continuous because the property is invariant under translation.
          – SmileyCraft
          2 days ago




          Roughly speaking, scalar multiplication is continuous exactly by the "algebraically openness" property, and vector addition is continuous because the property is invariant under translation.
          – SmileyCraft
          2 days ago












          @SimleyCraft I a m still not sure it's necessarily true (in anycase I understand how to solve the problem with the $phi - psi$ argument). For example why should scalar mult be continues? let $x_0 in X, lambda_0 in mathbb{R}, lambda_0 x_0 in U$ open, you would want a neighborhood of $x_0$, V, for which there exists $delta>0$, $vert lambda - lambda_0 vert < delta implies lambda V subset U$
          – pitariver
          2 days ago






          @SimleyCraft I a m still not sure it's necessarily true (in anycase I understand how to solve the problem with the $phi - psi$ argument). For example why should scalar mult be continues? let $x_0 in X, lambda_0 in mathbb{R}, lambda_0 x_0 in U$ open, you would want a neighborhood of $x_0$, V, for which there exists $delta>0$, $vert lambda - lambda_0 vert < delta implies lambda V subset U$
          – pitariver
          2 days ago














          @pitariver I must admit I underestimated the difficulty of proving this fact. I did already realize the problems you mentioned, but I had some ideas of proving this, however they did not work after all. I edited the post with a new answer. However I would still love to know whether the "algebraic openness" defines a linear topology.
          – SmileyCraft
          2 days ago




          @pitariver I must admit I underestimated the difficulty of proving this fact. I did already realize the problems you mentioned, but I had some ideas of proving this, however they did not work after all. I edited the post with a new answer. However I would still love to know whether the "algebraic openness" defines a linear topology.
          – SmileyCraft
          2 days ago












          Thanks for the link, though sadly, it still doesn't solve the problem: I asked about a strict separation, yet Theorem 4 in the paper only guarantees "weak" separation(which I already heard of). The counter example given immediately after the theorem also doesn't tell us much about this case. I am starting to wonder whether the statement is actually true, even for the "half strict" case.
          – pitariver
          yesterday






          Thanks for the link, though sadly, it still doesn't solve the problem: I asked about a strict separation, yet Theorem 4 in the paper only guarantees "weak" separation(which I already heard of). The counter example given immediately after the theorem also doesn't tell us much about this case. I am starting to wonder whether the statement is actually true, even for the "half strict" case.
          – pitariver
          yesterday














          Also in regards to the "algebraically open" sets defining a TVS, I am not entirely sure it's true. Thanks for the attention so far!
          – pitariver
          yesterday




          Also in regards to the "algebraically open" sets defining a TVS, I am not entirely sure it's true. Thanks for the attention so far!
          – pitariver
          yesterday


















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