On reflexivity properties of the Schwartz space
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Consider $mathscrS$ the Schwartz space of rapidly decreasing complex smooth functions over $mathbbR^d$, equipped with its usual metric topology, and $mathscrS'$ its topological dual (the space of tempered distributions). I have a few questions concerning the reflexivity of the space $mathscrS$ when $mathscrS'$ is equipped with the weak-$*$ topology.
The first question: Does it hold that $(mathscrS')' = mathscrS$ in an algebraic sense?. To be precise, if $L : mathscrS' to mathbbC$ is a linear functional, continuous with the weak-$*$ topology on $mathscrS'$, is there a unique function $varphi in mathscrS$ such that $$L(T) = langle T , varphi rangle, quad forall T in mathscrS'. $$
According to classical litterature in Theory of Distributions (for example, Shcwartz's book Théorie des Distributions), this is true when $mathscrS'$ is equipped with the strong topology, although I have trouble understanding the notion of this topology (for instance, what does it means for a subset of $mathscrS$ to be bounded? Is it just the classical definition on metric spaces?). I would like to know if it also holds for the weak-$*$ topology.
The second question is if the weak topology on $mathscrS$ is equivalent to the metric one. To be precise, if $(varphi_n)_n in mathbbN$ is a sequence of functions in $mathscrS$ such that $ langle T , varphi_n rangle to 0 $ for all $T in mathscrS'$, does it hold that $varphi_n to 0 $ in the sense of the metric of the Schwartz space?
Thank you very much for your help.
schwartz-space reflexive-space
asked Aug 7 at 19:42
CarrizoV
312
add a comment |Â
up vote
4
down vote
favorite
Consider $mathscrS$ the Schwartz space of rapidly decreasing complex smooth functions over $mathbbR^d$, equipped with its usual metric topology, and $mathscrS'$ its topological dual (the space of tempered distributions). I have a few questions concerning the reflexivity of the space $mathscrS$ when $mathscrS'$ is equipped with the weak-$*$ topology.
The first question: Does it hold that $(mathscrS')' = mathscrS$ in an algebraic sense?. To be precise, if $L : mathscrS' to mathbbC$ is a linear functional, continuous with the weak-$*$ topology on $mathscrS'$, is there a unique function $varphi in mathscrS$ such that $$L(T) = langle T , varphi rangle, quad forall T in mathscrS'. $$
According to classical litterature in Theory of Distributions (for example, Shcwartz's book Théorie des Distributions), this is true when $mathscrS'$ is equipped with the strong topology, although I have trouble understanding the notion of this topology (for instance, what does it means for a subset of $mathscrS$ to be bounded? Is it just the classical definition on metric spaces?). I would like to know if it also holds for the weak-$*$ topology.
The second question is if the weak topology on $mathscrS$ is equivalent to the metric one. To be precise, if $(varphi_n)_n in mathbbN$ is a sequence of functions in $mathscrS$ such that $ langle T , varphi_n rangle to 0 $ for all $T in mathscrS'$, does it hold that $varphi_n to 0 $ in the sense of the metric of the Schwartz space?
Thank you very much for your help.
schwartz-space reflexive-space
asked Aug 7 at 19:42
CarrizoV
312
For the first question the answer is yes because if the map $L$ is continuous for the weak-$ast$ topology on the domain, it will be continuous for the strong topology. The second question is interesting. I don't know the answer right off the bat. The best way to approach it, I think, is to use the isomorphism of Schwartz space with the space of sequences with fast decay, see e.g. aip.scitation.org/doi/abs/10.1063/1.1665472
– Abdelmalek Abdesselam
Aug 9 at 15:27
add a comment |Â
up vote
4
down vote
favorite
up vote
4
down vote
favorite
Consider $mathscrS$ the Schwartz space of rapidly decreasing complex smooth functions over $mathbbR^d$, equipped with its usual metric topology, and $mathscrS'$ its topological dual (the space of tempered distributions). I have a few questions concerning the reflexivity of the space $mathscrS$ when $mathscrS'$ is equipped with the weak-$*$ topology.
The first question: Does it hold that $(mathscrS')' = mathscrS$ in an algebraic sense?. To be precise, if $L : mathscrS' to mathbbC$ is a linear functional, continuous with the weak-$*$ topology on $mathscrS'$, is there a unique function $varphi in mathscrS$ such that $$L(T) = langle T , varphi rangle, quad forall T in mathscrS'. $$
According to classical litterature in Theory of Distributions (for example, Shcwartz's book Théorie des Distributions), this is true when $mathscrS'$ is equipped with the strong topology, although I have trouble understanding the notion of this topology (for instance, what does it means for a subset of $mathscrS$ to be bounded? Is it just the classical definition on metric spaces?). I would like to know if it also holds for the weak-$*$ topology.
The second question is if the weak topology on $mathscrS$ is equivalent to the metric one. To be precise, if $(varphi_n)_n in mathbbN$ is a sequence of functions in $mathscrS$ such that $ langle T , varphi_n rangle to 0 $ for all $T in mathscrS'$, does it hold that $varphi_n to 0 $ in the sense of the metric of the Schwartz space?
Thank you very much for your help.
schwartz-space reflexive-space
asked Aug 7 at 19:42
CarrizoV
312
Consider $mathscrS$ the Schwartz space of rapidly decreasing complex smooth functions over $mathbbR^d$, equipped with its usual metric topology, and $mathscrS'$ its topological dual (the space of tempered distributions). I have a few questions concerning the reflexivity of the space $mathscrS$ when $mathscrS'$ is equipped with the weak-$*$ topology.
The first question: Does it hold that $(mathscrS')' = mathscrS$ in an algebraic sense?. To be precise, if $L : mathscrS' to mathbbC$ is a linear functional, continuous with the weak-$*$ topology on $mathscrS'$, is there a unique function $varphi in mathscrS$ such that $$L(T) = langle T , varphi rangle, quad forall T in mathscrS'. $$
According to classical litterature in Theory of Distributions (for example, Shcwartz's book Théorie des Distributions), this is true when $mathscrS'$ is equipped with the strong topology, although I have trouble understanding the notion of this topology (for instance, what does it means for a subset of $mathscrS$ to be bounded? Is it just the classical definition on metric spaces?). I would like to know if it also holds for the weak-$*$ topology.
The second question is if the weak topology on $mathscrS$ is equivalent to the metric one. To be precise, if $(varphi_n)_n in mathbbN$ is a sequence of functions in $mathscrS$ such that $ langle T , varphi_n rangle to 0 $ for all $T in mathscrS'$, does it hold that $varphi_n to 0 $ in the sense of the metric of the Schwartz space?
Thank you very much for your help.
schwartz-space reflexive-space
asked Aug 7 at 19:42
CarrizoV
312
asked Aug 7 at 19:42
CarrizoV
312
asked Aug 7 at 19:42
CarrizoV
312
asked Aug 7 at 19:42
CarrizoV
312
312
For the first question the answer is yes because if the map $L$ is continuous for the weak-$ast$ topology on the domain, it will be continuous for the strong topology. The second question is interesting. I don't know the answer right off the bat. The best way to approach it, I think, is to use the isomorphism of Schwartz space with the space of sequences with fast decay, see e.g. aip.scitation.org/doi/abs/10.1063/1.1665472
– Abdelmalek Abdesselam
Aug 9 at 15:27
add a comment |Â
For the first question the answer is yes because if the map $L$ is continuous for the weak-$ast$ topology on the domain, it will be continuous for the strong topology. The second question is interesting. I don't know the answer right off the bat. The best way to approach it, I think, is to use the isomorphism of Schwartz space with the space of sequences with fast decay, see e.g. aip.scitation.org/doi/abs/10.1063/1.1665472
– Abdelmalek Abdesselam
Aug 9 at 15:27
For the first question the answer is yes because if the map $L$ is continuous for the weak-$ast$ topology on the domain, it will be continuous for the strong topology. The second question is interesting. I don't know the answer right off the bat. The best way to approach it, I think, is to use the isomorphism of Schwartz space with the space of sequences with fast decay, see e.g. aip.scitation.org/doi/abs/10.1063/1.1665472
– Abdelmalek Abdesselam
Aug 9 at 15:27
For the first question the answer is yes because if the map $L$ is continuous for the weak-$ast$ topology on the domain, it will be continuous for the strong topology. The second question is interesting. I don't know the answer right off the bat. The best way to approach it, I think, is to use the isomorphism of Schwartz space with the space of sequences with fast decay, see e.g. aip.scitation.org/doi/abs/10.1063/1.1665472
– Abdelmalek Abdesselam
Aug 9 at 15:27
add a comment |Â
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For the first question the answer is yes because if the map $L$ is continuous for the weak-$ast$ topology on the domain, it will be continuous for the strong topology. The second question is interesting. I don't know the answer right off the bat. The best way to approach it, I think, is to use the isomorphism of Schwartz space with the space of sequences with fast decay, see e.g. aip.scitation.org/doi/abs/10.1063/1.1665472
– Abdelmalek Abdesselam
Aug 9 at 15:27