Weak-$*$ topology on algebraic dual$C(K)^*$ in the weak*-topologyOn the Banach–Alaoglu theorem: is the unit ball of an equivalent norm also weak-* compact?Is this statement true? (characterize elements of dual group)Understanding Wikipedia's definition for latticeIs the Haar measure on Abelian groups regular?Why is the space of finite Borel-measure dual to the space of finite continuous function.Fourier Transform of Measures on Hilbert, Banach, or general Topological Vector SpaceHaar Measures on SolenoidsBound on the measure of a sum of sets$sigma$ finiteness of locally compact groups with haar measure

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Weak-$*$ topology on algebraic dual


$C(K)^*$ in the weak*-topologyOn the Banach–Alaoglu theorem: is the unit ball of an equivalent norm also weak-* compact?Is this statement true? (characterize elements of dual group)Understanding Wikipedia's definition for latticeIs the Haar measure on Abelian groups regular?Why is the space of finite Borel-measure dual to the space of finite continuous function.Fourier Transform of Measures on Hilbert, Banach, or general Topological Vector SpaceHaar Measures on SolenoidsBound on the measure of a sum of sets$sigma$ finiteness of locally compact groups with haar measure













4












$begingroup$


I was looking at




Izzo, Alexander J., A functional analysis proof of the existence of Haar measure on locally compact Abelian groups, Proc. Am. Math. Soc. 115, No. 2, 581-583 (1992). ZBL0777.28006.




which proves existence of the Haar-measure for locally compact abelian groups using the Markov-Kakutani theorem.



What I find strange is that the Haar measure is constructed as an element of the dual of $C_c(X)$. But for noncompact $X$ (such as $X$ being the real numbers $Bbb R$) this must be an unbounded functional (as the Lebesgue-measure on $Bbb R$ is not finite). It seems like the author has no problem with this, and (without mentioning it further) goes on to define a weak-* topology for this case and even uses Banach-Alaoglu.



I have not seen this being done this way before, am I misunderstanding something or can one define a weak-* topology on the algebraic dual of a TVS without any problems?










share|cite|improve this question











$endgroup$
















    4












    $begingroup$


    I was looking at




    Izzo, Alexander J., A functional analysis proof of the existence of Haar measure on locally compact Abelian groups, Proc. Am. Math. Soc. 115, No. 2, 581-583 (1992). ZBL0777.28006.




    which proves existence of the Haar-measure for locally compact abelian groups using the Markov-Kakutani theorem.



    What I find strange is that the Haar measure is constructed as an element of the dual of $C_c(X)$. But for noncompact $X$ (such as $X$ being the real numbers $Bbb R$) this must be an unbounded functional (as the Lebesgue-measure on $Bbb R$ is not finite). It seems like the author has no problem with this, and (without mentioning it further) goes on to define a weak-* topology for this case and even uses Banach-Alaoglu.



    I have not seen this being done this way before, am I misunderstanding something or can one define a weak-* topology on the algebraic dual of a TVS without any problems?










    share|cite|improve this question











    $endgroup$














      4












      4








      4





      $begingroup$


      I was looking at




      Izzo, Alexander J., A functional analysis proof of the existence of Haar measure on locally compact Abelian groups, Proc. Am. Math. Soc. 115, No. 2, 581-583 (1992). ZBL0777.28006.




      which proves existence of the Haar-measure for locally compact abelian groups using the Markov-Kakutani theorem.



      What I find strange is that the Haar measure is constructed as an element of the dual of $C_c(X)$. But for noncompact $X$ (such as $X$ being the real numbers $Bbb R$) this must be an unbounded functional (as the Lebesgue-measure on $Bbb R$ is not finite). It seems like the author has no problem with this, and (without mentioning it further) goes on to define a weak-* topology for this case and even uses Banach-Alaoglu.



      I have not seen this being done this way before, am I misunderstanding something or can one define a weak-* topology on the algebraic dual of a TVS without any problems?










      share|cite|improve this question











      $endgroup$




      I was looking at




      Izzo, Alexander J., A functional analysis proof of the existence of Haar measure on locally compact Abelian groups, Proc. Am. Math. Soc. 115, No. 2, 581-583 (1992). ZBL0777.28006.




      which proves existence of the Haar-measure for locally compact abelian groups using the Markov-Kakutani theorem.



      What I find strange is that the Haar measure is constructed as an element of the dual of $C_c(X)$. But for noncompact $X$ (such as $X$ being the real numbers $Bbb R$) this must be an unbounded functional (as the Lebesgue-measure on $Bbb R$ is not finite). It seems like the author has no problem with this, and (without mentioning it further) goes on to define a weak-* topology for this case and even uses Banach-Alaoglu.



      I have not seen this being done this way before, am I misunderstanding something or can one define a weak-* topology on the algebraic dual of a TVS without any problems?







      functional-analysis harmonic-analysis haar-measure






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Feb 14 at 13:56









      Asaf Karagila

      306k33437768




      306k33437768










      asked Feb 14 at 13:47









      Nathanael SchillingNathanael Schilling

      1249




      1249




















          3 Answers
          3






          active

          oldest

          votes


















          1












          $begingroup$

          I agree with Paul Garrett that it seems more reasonable to endow $C_c(G)$ with the strict inductive limit topology and consider the topological dual, which is isomorphic to the space of all Radon measures on $G$ (in particular, infinite measures also give rise to continuous functionals).



          However, the linked lemma also holds for the algebraic dual $X^ast$. What the author calls the weak$^ast$ topology on $X^ast$ is the $sigma(X^ast,X)$ topology, i.e, the locally convex topology generated by the seminorms $p_xcolonphimapsto|phi(x)|$ for $xin X$. If you view $X^ast$ as a subspace of $mathbbR^X$, then this is just the (induced topology of) the product topology. This already gives you a hint how to prove this generalized Banach-Alaoglu theorem.



          For $xin X$ let $K_x$ be the closure of $Lambda(x)midLambdain K$. By assumption, $K_x$ is compact. Then $K$ is a subset of $prod_xin XK_x$, which is compact by Tychonoff's theorem. Since $K$ is also assumed to be closed in $X^ast$, it suffices to show that $X^ast$ is a closed subset of $mathbbR^X$, which is pretty straightforward.






          share|cite|improve this answer









          $endgroup$




















            1












            $begingroup$

            The intent of the quoted lemma in Izzo's paper is to show the compactness of a certain subset of linear functionals in the weak* topology, and I agree with @MaoWao that the linked lemma also holds for the algebraic dual $X^*$. The definition of the weak* topology on a space $X^*$ of functionals or generally functions on a domain space $X$ does not depend on any topology on $X$, but rather only on the topology of the value field (e.g. $R$ or $C$), and if a theorem like Banach-Alaoglu is desired, then on the local compactness of the value field.



            It is not the 1st time I see the weak* topology on the space of $all$ linear functionals, i.e. on the algebraic dual: on page 108 of John L. Kelley's famous Topology book there is exercise W on "Functionals on real linear spaces" which starts like follows:



            "Let $(X, +,.)$ be a real linear space. A real-valued linear function on $X$ is called a $linear functional$. The set $Z$ of all linear functionals on $X$ is, with the natural definition of addition and scalar multiplication, a real linear space. It is clear that $Z$ is a subset of the product $R^X$, where $R$ is the set of real numbers. The relativized product topology for $Z$ is called the $weak^*$ or $w^*$ topology (the $simple$ topology)."



            Kelley then moved on to formulate a $Density$ $lemma$ and an $Evaluation$ $Theorem$ as exercises with respect to the weak* topology on the Algebraic dual $Z$ of $X$.






            share|cite|improve this answer









            $endgroup$




















              0












              $begingroup$

              Probably the topology on $C^o_c(mathbb R)$ (or for other non-compact topological group in place of $mathbb R$) is not what you anticipated. It is not sup norm, for example. It is an ascending union, categorically a "strict colimit" (=strict inductive limit=...) of spaces of the form $C^o(K)$ as $K$ ranges over compact subsets of $mathbb R$. (These are Banach spaces.)



              In particular, from the characterizing mapping property of "colimit", a linear map or functional from $C^o_c(mathbb R)$ is continuous if and only if its restriction to each $C^o(K)$ is continuous.



              So the dual of $C^o_c(mathbb R)$ does not include "unbounded" functionals in a true sense, because even on such spaces "continuous" is still equivalent to "bounded"... but "bounded" has a more complicated sense.






              share|cite|improve this answer









              $endgroup$












              • $begingroup$
                Sure, if I don't take C_c( R ) with the norm topology but with some different topology then I can get more continuous functionals. But this doesn't answer my question: in Izzo's proof (if I understand it correctly) he explicitly takes the space of all linear functionals on C_c and gives it a topology and claims that something like Banach-Alaouglo holds.
                $endgroup$
                – Nathanael Schilling
                Feb 14 at 22:37










              • $begingroup$
                @NathanaelSchilling, I would strongly doubt that the intent is to take all linear functionals on $C^o_c(mathbb R)$, since things would run amok. Ah... Just realized: when functional analysts say "dual" they almost-surely mean "continuous dual". Rarely, if ever, "purely-algebraic dual". I'd bet a dollar that that's the intent.
                $endgroup$
                – paul garrett
                Feb 14 at 22:44











              • $begingroup$
                Yes usually I would agree. I've never seen the algebraic dual being used in functional analysis. Except (implicitly) in the Riesz-Markov theorem, which takes an arbitrary positive functional on C_c and shows it can be represented by a measure. But since the author emphasises the word "all" here and the context is very similar, it got me wondering if this perhaps is possible... I don't see any obvious obstructions to defining a weak^* topology on this space, and even if the result is not a topological vector space it could be that some sort of compactness results hold even in this setting?
                $endgroup$
                – Nathanael Schilling
                Feb 15 at 14:19










              • $begingroup$
                Also, btw, the "positive functional" version of Riesz-Markov-Kakutani is just a veiled form of a continuity assumption (Banach-Steinhaus...), but/and has a certain popularity because it's easier to say "positive" than to describe the topology on compactly-supported continuous functions.
                $endgroup$
                – paul garrett
                Feb 16 at 18:26










              Your Answer





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              3 Answers
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              active

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              3 Answers
              3






              active

              oldest

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              active

              oldest

              votes






              active

              oldest

              votes









              1












              $begingroup$

              I agree with Paul Garrett that it seems more reasonable to endow $C_c(G)$ with the strict inductive limit topology and consider the topological dual, which is isomorphic to the space of all Radon measures on $G$ (in particular, infinite measures also give rise to continuous functionals).



              However, the linked lemma also holds for the algebraic dual $X^ast$. What the author calls the weak$^ast$ topology on $X^ast$ is the $sigma(X^ast,X)$ topology, i.e, the locally convex topology generated by the seminorms $p_xcolonphimapsto|phi(x)|$ for $xin X$. If you view $X^ast$ as a subspace of $mathbbR^X$, then this is just the (induced topology of) the product topology. This already gives you a hint how to prove this generalized Banach-Alaoglu theorem.



              For $xin X$ let $K_x$ be the closure of $Lambda(x)midLambdain K$. By assumption, $K_x$ is compact. Then $K$ is a subset of $prod_xin XK_x$, which is compact by Tychonoff's theorem. Since $K$ is also assumed to be closed in $X^ast$, it suffices to show that $X^ast$ is a closed subset of $mathbbR^X$, which is pretty straightforward.






              share|cite|improve this answer









              $endgroup$

















                1












                $begingroup$

                I agree with Paul Garrett that it seems more reasonable to endow $C_c(G)$ with the strict inductive limit topology and consider the topological dual, which is isomorphic to the space of all Radon measures on $G$ (in particular, infinite measures also give rise to continuous functionals).



                However, the linked lemma also holds for the algebraic dual $X^ast$. What the author calls the weak$^ast$ topology on $X^ast$ is the $sigma(X^ast,X)$ topology, i.e, the locally convex topology generated by the seminorms $p_xcolonphimapsto|phi(x)|$ for $xin X$. If you view $X^ast$ as a subspace of $mathbbR^X$, then this is just the (induced topology of) the product topology. This already gives you a hint how to prove this generalized Banach-Alaoglu theorem.



                For $xin X$ let $K_x$ be the closure of $Lambda(x)midLambdain K$. By assumption, $K_x$ is compact. Then $K$ is a subset of $prod_xin XK_x$, which is compact by Tychonoff's theorem. Since $K$ is also assumed to be closed in $X^ast$, it suffices to show that $X^ast$ is a closed subset of $mathbbR^X$, which is pretty straightforward.






                share|cite|improve this answer









                $endgroup$















                  1












                  1








                  1





                  $begingroup$

                  I agree with Paul Garrett that it seems more reasonable to endow $C_c(G)$ with the strict inductive limit topology and consider the topological dual, which is isomorphic to the space of all Radon measures on $G$ (in particular, infinite measures also give rise to continuous functionals).



                  However, the linked lemma also holds for the algebraic dual $X^ast$. What the author calls the weak$^ast$ topology on $X^ast$ is the $sigma(X^ast,X)$ topology, i.e, the locally convex topology generated by the seminorms $p_xcolonphimapsto|phi(x)|$ for $xin X$. If you view $X^ast$ as a subspace of $mathbbR^X$, then this is just the (induced topology of) the product topology. This already gives you a hint how to prove this generalized Banach-Alaoglu theorem.



                  For $xin X$ let $K_x$ be the closure of $Lambda(x)midLambdain K$. By assumption, $K_x$ is compact. Then $K$ is a subset of $prod_xin XK_x$, which is compact by Tychonoff's theorem. Since $K$ is also assumed to be closed in $X^ast$, it suffices to show that $X^ast$ is a closed subset of $mathbbR^X$, which is pretty straightforward.






                  share|cite|improve this answer









                  $endgroup$



                  I agree with Paul Garrett that it seems more reasonable to endow $C_c(G)$ with the strict inductive limit topology and consider the topological dual, which is isomorphic to the space of all Radon measures on $G$ (in particular, infinite measures also give rise to continuous functionals).



                  However, the linked lemma also holds for the algebraic dual $X^ast$. What the author calls the weak$^ast$ topology on $X^ast$ is the $sigma(X^ast,X)$ topology, i.e, the locally convex topology generated by the seminorms $p_xcolonphimapsto|phi(x)|$ for $xin X$. If you view $X^ast$ as a subspace of $mathbbR^X$, then this is just the (induced topology of) the product topology. This already gives you a hint how to prove this generalized Banach-Alaoglu theorem.



                  For $xin X$ let $K_x$ be the closure of $Lambda(x)midLambdain K$. By assumption, $K_x$ is compact. Then $K$ is a subset of $prod_xin XK_x$, which is compact by Tychonoff's theorem. Since $K$ is also assumed to be closed in $X^ast$, it suffices to show that $X^ast$ is a closed subset of $mathbbR^X$, which is pretty straightforward.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Feb 18 at 9:52









                  MaoWaoMaoWao

                  3,783617




                  3,783617





















                      1












                      $begingroup$

                      The intent of the quoted lemma in Izzo's paper is to show the compactness of a certain subset of linear functionals in the weak* topology, and I agree with @MaoWao that the linked lemma also holds for the algebraic dual $X^*$. The definition of the weak* topology on a space $X^*$ of functionals or generally functions on a domain space $X$ does not depend on any topology on $X$, but rather only on the topology of the value field (e.g. $R$ or $C$), and if a theorem like Banach-Alaoglu is desired, then on the local compactness of the value field.



                      It is not the 1st time I see the weak* topology on the space of $all$ linear functionals, i.e. on the algebraic dual: on page 108 of John L. Kelley's famous Topology book there is exercise W on "Functionals on real linear spaces" which starts like follows:



                      "Let $(X, +,.)$ be a real linear space. A real-valued linear function on $X$ is called a $linear functional$. The set $Z$ of all linear functionals on $X$ is, with the natural definition of addition and scalar multiplication, a real linear space. It is clear that $Z$ is a subset of the product $R^X$, where $R$ is the set of real numbers. The relativized product topology for $Z$ is called the $weak^*$ or $w^*$ topology (the $simple$ topology)."



                      Kelley then moved on to formulate a $Density$ $lemma$ and an $Evaluation$ $Theorem$ as exercises with respect to the weak* topology on the Algebraic dual $Z$ of $X$.






                      share|cite|improve this answer









                      $endgroup$

















                        1












                        $begingroup$

                        The intent of the quoted lemma in Izzo's paper is to show the compactness of a certain subset of linear functionals in the weak* topology, and I agree with @MaoWao that the linked lemma also holds for the algebraic dual $X^*$. The definition of the weak* topology on a space $X^*$ of functionals or generally functions on a domain space $X$ does not depend on any topology on $X$, but rather only on the topology of the value field (e.g. $R$ or $C$), and if a theorem like Banach-Alaoglu is desired, then on the local compactness of the value field.



                        It is not the 1st time I see the weak* topology on the space of $all$ linear functionals, i.e. on the algebraic dual: on page 108 of John L. Kelley's famous Topology book there is exercise W on "Functionals on real linear spaces" which starts like follows:



                        "Let $(X, +,.)$ be a real linear space. A real-valued linear function on $X$ is called a $linear functional$. The set $Z$ of all linear functionals on $X$ is, with the natural definition of addition and scalar multiplication, a real linear space. It is clear that $Z$ is a subset of the product $R^X$, where $R$ is the set of real numbers. The relativized product topology for $Z$ is called the $weak^*$ or $w^*$ topology (the $simple$ topology)."



                        Kelley then moved on to formulate a $Density$ $lemma$ and an $Evaluation$ $Theorem$ as exercises with respect to the weak* topology on the Algebraic dual $Z$ of $X$.






                        share|cite|improve this answer









                        $endgroup$















                          1












                          1








                          1





                          $begingroup$

                          The intent of the quoted lemma in Izzo's paper is to show the compactness of a certain subset of linear functionals in the weak* topology, and I agree with @MaoWao that the linked lemma also holds for the algebraic dual $X^*$. The definition of the weak* topology on a space $X^*$ of functionals or generally functions on a domain space $X$ does not depend on any topology on $X$, but rather only on the topology of the value field (e.g. $R$ or $C$), and if a theorem like Banach-Alaoglu is desired, then on the local compactness of the value field.



                          It is not the 1st time I see the weak* topology on the space of $all$ linear functionals, i.e. on the algebraic dual: on page 108 of John L. Kelley's famous Topology book there is exercise W on "Functionals on real linear spaces" which starts like follows:



                          "Let $(X, +,.)$ be a real linear space. A real-valued linear function on $X$ is called a $linear functional$. The set $Z$ of all linear functionals on $X$ is, with the natural definition of addition and scalar multiplication, a real linear space. It is clear that $Z$ is a subset of the product $R^X$, where $R$ is the set of real numbers. The relativized product topology for $Z$ is called the $weak^*$ or $w^*$ topology (the $simple$ topology)."



                          Kelley then moved on to formulate a $Density$ $lemma$ and an $Evaluation$ $Theorem$ as exercises with respect to the weak* topology on the Algebraic dual $Z$ of $X$.






                          share|cite|improve this answer









                          $endgroup$



                          The intent of the quoted lemma in Izzo's paper is to show the compactness of a certain subset of linear functionals in the weak* topology, and I agree with @MaoWao that the linked lemma also holds for the algebraic dual $X^*$. The definition of the weak* topology on a space $X^*$ of functionals or generally functions on a domain space $X$ does not depend on any topology on $X$, but rather only on the topology of the value field (e.g. $R$ or $C$), and if a theorem like Banach-Alaoglu is desired, then on the local compactness of the value field.



                          It is not the 1st time I see the weak* topology on the space of $all$ linear functionals, i.e. on the algebraic dual: on page 108 of John L. Kelley's famous Topology book there is exercise W on "Functionals on real linear spaces" which starts like follows:



                          "Let $(X, +,.)$ be a real linear space. A real-valued linear function on $X$ is called a $linear functional$. The set $Z$ of all linear functionals on $X$ is, with the natural definition of addition and scalar multiplication, a real linear space. It is clear that $Z$ is a subset of the product $R^X$, where $R$ is the set of real numbers. The relativized product topology for $Z$ is called the $weak^*$ or $w^*$ topology (the $simple$ topology)."



                          Kelley then moved on to formulate a $Density$ $lemma$ and an $Evaluation$ $Theorem$ as exercises with respect to the weak* topology on the Algebraic dual $Z$ of $X$.







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered 17 hours ago









                          Matthias HübnerMatthias Hübner

                          484




                          484





















                              0












                              $begingroup$

                              Probably the topology on $C^o_c(mathbb R)$ (or for other non-compact topological group in place of $mathbb R$) is not what you anticipated. It is not sup norm, for example. It is an ascending union, categorically a "strict colimit" (=strict inductive limit=...) of spaces of the form $C^o(K)$ as $K$ ranges over compact subsets of $mathbb R$. (These are Banach spaces.)



                              In particular, from the characterizing mapping property of "colimit", a linear map or functional from $C^o_c(mathbb R)$ is continuous if and only if its restriction to each $C^o(K)$ is continuous.



                              So the dual of $C^o_c(mathbb R)$ does not include "unbounded" functionals in a true sense, because even on such spaces "continuous" is still equivalent to "bounded"... but "bounded" has a more complicated sense.






                              share|cite|improve this answer









                              $endgroup$












                              • $begingroup$
                                Sure, if I don't take C_c( R ) with the norm topology but with some different topology then I can get more continuous functionals. But this doesn't answer my question: in Izzo's proof (if I understand it correctly) he explicitly takes the space of all linear functionals on C_c and gives it a topology and claims that something like Banach-Alaouglo holds.
                                $endgroup$
                                – Nathanael Schilling
                                Feb 14 at 22:37










                              • $begingroup$
                                @NathanaelSchilling, I would strongly doubt that the intent is to take all linear functionals on $C^o_c(mathbb R)$, since things would run amok. Ah... Just realized: when functional analysts say "dual" they almost-surely mean "continuous dual". Rarely, if ever, "purely-algebraic dual". I'd bet a dollar that that's the intent.
                                $endgroup$
                                – paul garrett
                                Feb 14 at 22:44











                              • $begingroup$
                                Yes usually I would agree. I've never seen the algebraic dual being used in functional analysis. Except (implicitly) in the Riesz-Markov theorem, which takes an arbitrary positive functional on C_c and shows it can be represented by a measure. But since the author emphasises the word "all" here and the context is very similar, it got me wondering if this perhaps is possible... I don't see any obvious obstructions to defining a weak^* topology on this space, and even if the result is not a topological vector space it could be that some sort of compactness results hold even in this setting?
                                $endgroup$
                                – Nathanael Schilling
                                Feb 15 at 14:19










                              • $begingroup$
                                Also, btw, the "positive functional" version of Riesz-Markov-Kakutani is just a veiled form of a continuity assumption (Banach-Steinhaus...), but/and has a certain popularity because it's easier to say "positive" than to describe the topology on compactly-supported continuous functions.
                                $endgroup$
                                – paul garrett
                                Feb 16 at 18:26















                              0












                              $begingroup$

                              Probably the topology on $C^o_c(mathbb R)$ (or for other non-compact topological group in place of $mathbb R$) is not what you anticipated. It is not sup norm, for example. It is an ascending union, categorically a "strict colimit" (=strict inductive limit=...) of spaces of the form $C^o(K)$ as $K$ ranges over compact subsets of $mathbb R$. (These are Banach spaces.)



                              In particular, from the characterizing mapping property of "colimit", a linear map or functional from $C^o_c(mathbb R)$ is continuous if and only if its restriction to each $C^o(K)$ is continuous.



                              So the dual of $C^o_c(mathbb R)$ does not include "unbounded" functionals in a true sense, because even on such spaces "continuous" is still equivalent to "bounded"... but "bounded" has a more complicated sense.






                              share|cite|improve this answer









                              $endgroup$












                              • $begingroup$
                                Sure, if I don't take C_c( R ) with the norm topology but with some different topology then I can get more continuous functionals. But this doesn't answer my question: in Izzo's proof (if I understand it correctly) he explicitly takes the space of all linear functionals on C_c and gives it a topology and claims that something like Banach-Alaouglo holds.
                                $endgroup$
                                – Nathanael Schilling
                                Feb 14 at 22:37










                              • $begingroup$
                                @NathanaelSchilling, I would strongly doubt that the intent is to take all linear functionals on $C^o_c(mathbb R)$, since things would run amok. Ah... Just realized: when functional analysts say "dual" they almost-surely mean "continuous dual". Rarely, if ever, "purely-algebraic dual". I'd bet a dollar that that's the intent.
                                $endgroup$
                                – paul garrett
                                Feb 14 at 22:44











                              • $begingroup$
                                Yes usually I would agree. I've never seen the algebraic dual being used in functional analysis. Except (implicitly) in the Riesz-Markov theorem, which takes an arbitrary positive functional on C_c and shows it can be represented by a measure. But since the author emphasises the word "all" here and the context is very similar, it got me wondering if this perhaps is possible... I don't see any obvious obstructions to defining a weak^* topology on this space, and even if the result is not a topological vector space it could be that some sort of compactness results hold even in this setting?
                                $endgroup$
                                – Nathanael Schilling
                                Feb 15 at 14:19










                              • $begingroup$
                                Also, btw, the "positive functional" version of Riesz-Markov-Kakutani is just a veiled form of a continuity assumption (Banach-Steinhaus...), but/and has a certain popularity because it's easier to say "positive" than to describe the topology on compactly-supported continuous functions.
                                $endgroup$
                                – paul garrett
                                Feb 16 at 18:26













                              0












                              0








                              0





                              $begingroup$

                              Probably the topology on $C^o_c(mathbb R)$ (or for other non-compact topological group in place of $mathbb R$) is not what you anticipated. It is not sup norm, for example. It is an ascending union, categorically a "strict colimit" (=strict inductive limit=...) of spaces of the form $C^o(K)$ as $K$ ranges over compact subsets of $mathbb R$. (These are Banach spaces.)



                              In particular, from the characterizing mapping property of "colimit", a linear map or functional from $C^o_c(mathbb R)$ is continuous if and only if its restriction to each $C^o(K)$ is continuous.



                              So the dual of $C^o_c(mathbb R)$ does not include "unbounded" functionals in a true sense, because even on such spaces "continuous" is still equivalent to "bounded"... but "bounded" has a more complicated sense.






                              share|cite|improve this answer









                              $endgroup$



                              Probably the topology on $C^o_c(mathbb R)$ (or for other non-compact topological group in place of $mathbb R$) is not what you anticipated. It is not sup norm, for example. It is an ascending union, categorically a "strict colimit" (=strict inductive limit=...) of spaces of the form $C^o(K)$ as $K$ ranges over compact subsets of $mathbb R$. (These are Banach spaces.)



                              In particular, from the characterizing mapping property of "colimit", a linear map or functional from $C^o_c(mathbb R)$ is continuous if and only if its restriction to each $C^o(K)$ is continuous.



                              So the dual of $C^o_c(mathbb R)$ does not include "unbounded" functionals in a true sense, because even on such spaces "continuous" is still equivalent to "bounded"... but "bounded" has a more complicated sense.







                              share|cite|improve this answer












                              share|cite|improve this answer



                              share|cite|improve this answer










                              answered Feb 14 at 18:05









                              paul garrettpaul garrett

                              32k362118




                              32k362118











                              • $begingroup$
                                Sure, if I don't take C_c( R ) with the norm topology but with some different topology then I can get more continuous functionals. But this doesn't answer my question: in Izzo's proof (if I understand it correctly) he explicitly takes the space of all linear functionals on C_c and gives it a topology and claims that something like Banach-Alaouglo holds.
                                $endgroup$
                                – Nathanael Schilling
                                Feb 14 at 22:37










                              • $begingroup$
                                @NathanaelSchilling, I would strongly doubt that the intent is to take all linear functionals on $C^o_c(mathbb R)$, since things would run amok. Ah... Just realized: when functional analysts say "dual" they almost-surely mean "continuous dual". Rarely, if ever, "purely-algebraic dual". I'd bet a dollar that that's the intent.
                                $endgroup$
                                – paul garrett
                                Feb 14 at 22:44











                              • $begingroup$
                                Yes usually I would agree. I've never seen the algebraic dual being used in functional analysis. Except (implicitly) in the Riesz-Markov theorem, which takes an arbitrary positive functional on C_c and shows it can be represented by a measure. But since the author emphasises the word "all" here and the context is very similar, it got me wondering if this perhaps is possible... I don't see any obvious obstructions to defining a weak^* topology on this space, and even if the result is not a topological vector space it could be that some sort of compactness results hold even in this setting?
                                $endgroup$
                                – Nathanael Schilling
                                Feb 15 at 14:19










                              • $begingroup$
                                Also, btw, the "positive functional" version of Riesz-Markov-Kakutani is just a veiled form of a continuity assumption (Banach-Steinhaus...), but/and has a certain popularity because it's easier to say "positive" than to describe the topology on compactly-supported continuous functions.
                                $endgroup$
                                – paul garrett
                                Feb 16 at 18:26
















                              • $begingroup$
                                Sure, if I don't take C_c( R ) with the norm topology but with some different topology then I can get more continuous functionals. But this doesn't answer my question: in Izzo's proof (if I understand it correctly) he explicitly takes the space of all linear functionals on C_c and gives it a topology and claims that something like Banach-Alaouglo holds.
                                $endgroup$
                                – Nathanael Schilling
                                Feb 14 at 22:37










                              • $begingroup$
                                @NathanaelSchilling, I would strongly doubt that the intent is to take all linear functionals on $C^o_c(mathbb R)$, since things would run amok. Ah... Just realized: when functional analysts say "dual" they almost-surely mean "continuous dual". Rarely, if ever, "purely-algebraic dual". I'd bet a dollar that that's the intent.
                                $endgroup$
                                – paul garrett
                                Feb 14 at 22:44











                              • $begingroup$
                                Yes usually I would agree. I've never seen the algebraic dual being used in functional analysis. Except (implicitly) in the Riesz-Markov theorem, which takes an arbitrary positive functional on C_c and shows it can be represented by a measure. But since the author emphasises the word "all" here and the context is very similar, it got me wondering if this perhaps is possible... I don't see any obvious obstructions to defining a weak^* topology on this space, and even if the result is not a topological vector space it could be that some sort of compactness results hold even in this setting?
                                $endgroup$
                                – Nathanael Schilling
                                Feb 15 at 14:19










                              • $begingroup$
                                Also, btw, the "positive functional" version of Riesz-Markov-Kakutani is just a veiled form of a continuity assumption (Banach-Steinhaus...), but/and has a certain popularity because it's easier to say "positive" than to describe the topology on compactly-supported continuous functions.
                                $endgroup$
                                – paul garrett
                                Feb 16 at 18:26















                              $begingroup$
                              Sure, if I don't take C_c( R ) with the norm topology but with some different topology then I can get more continuous functionals. But this doesn't answer my question: in Izzo's proof (if I understand it correctly) he explicitly takes the space of all linear functionals on C_c and gives it a topology and claims that something like Banach-Alaouglo holds.
                              $endgroup$
                              – Nathanael Schilling
                              Feb 14 at 22:37




                              $begingroup$
                              Sure, if I don't take C_c( R ) with the norm topology but with some different topology then I can get more continuous functionals. But this doesn't answer my question: in Izzo's proof (if I understand it correctly) he explicitly takes the space of all linear functionals on C_c and gives it a topology and claims that something like Banach-Alaouglo holds.
                              $endgroup$
                              – Nathanael Schilling
                              Feb 14 at 22:37












                              $begingroup$
                              @NathanaelSchilling, I would strongly doubt that the intent is to take all linear functionals on $C^o_c(mathbb R)$, since things would run amok. Ah... Just realized: when functional analysts say "dual" they almost-surely mean "continuous dual". Rarely, if ever, "purely-algebraic dual". I'd bet a dollar that that's the intent.
                              $endgroup$
                              – paul garrett
                              Feb 14 at 22:44





                              $begingroup$
                              @NathanaelSchilling, I would strongly doubt that the intent is to take all linear functionals on $C^o_c(mathbb R)$, since things would run amok. Ah... Just realized: when functional analysts say "dual" they almost-surely mean "continuous dual". Rarely, if ever, "purely-algebraic dual". I'd bet a dollar that that's the intent.
                              $endgroup$
                              – paul garrett
                              Feb 14 at 22:44













                              $begingroup$
                              Yes usually I would agree. I've never seen the algebraic dual being used in functional analysis. Except (implicitly) in the Riesz-Markov theorem, which takes an arbitrary positive functional on C_c and shows it can be represented by a measure. But since the author emphasises the word "all" here and the context is very similar, it got me wondering if this perhaps is possible... I don't see any obvious obstructions to defining a weak^* topology on this space, and even if the result is not a topological vector space it could be that some sort of compactness results hold even in this setting?
                              $endgroup$
                              – Nathanael Schilling
                              Feb 15 at 14:19




                              $begingroup$
                              Yes usually I would agree. I've never seen the algebraic dual being used in functional analysis. Except (implicitly) in the Riesz-Markov theorem, which takes an arbitrary positive functional on C_c and shows it can be represented by a measure. But since the author emphasises the word "all" here and the context is very similar, it got me wondering if this perhaps is possible... I don't see any obvious obstructions to defining a weak^* topology on this space, and even if the result is not a topological vector space it could be that some sort of compactness results hold even in this setting?
                              $endgroup$
                              – Nathanael Schilling
                              Feb 15 at 14:19












                              $begingroup$
                              Also, btw, the "positive functional" version of Riesz-Markov-Kakutani is just a veiled form of a continuity assumption (Banach-Steinhaus...), but/and has a certain popularity because it's easier to say "positive" than to describe the topology on compactly-supported continuous functions.
                              $endgroup$
                              – paul garrett
                              Feb 16 at 18:26




                              $begingroup$
                              Also, btw, the "positive functional" version of Riesz-Markov-Kakutani is just a veiled form of a continuity assumption (Banach-Steinhaus...), but/and has a certain popularity because it's easier to say "positive" than to describe the topology on compactly-supported continuous functions.
                              $endgroup$
                              – paul garrett
                              Feb 16 at 18:26

















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