Is the dual $X^*$ of a metrizable topological vector space $X$ metrizable? [on hold]
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I edited my question including sufficient details.
Let $X$ be a metrizable topological vector space and $X^*$ be its dual space. Let $C$ be a collection of all bounded subsets of $X$, the topology $T$ on $X^*$ is generated by the family of seminorms of the form:
$$||phi||_A=sup_{xin A}|phi(x)|$$
where $phi$ is a continuous linear functional on $V$ and $A$ runs over the class $C$. Under what condition the dual $X^*$ is metrizable?
general-topology topological-vector-spaces
asked 2 days ago
Infinity
580313
put on hold as off-topic by user302797, Cesareo, Brahadeesh, Omran Kouba, Did yesterday
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – user302797, Cesareo, Brahadeesh, Omran Kouba, Did
If this question can be reworded to fit the rules in the help center, please edit the question.
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I edited my question including sufficient details.
Let $X$ be a metrizable topological vector space and $X^*$ be its dual space. Let $C$ be a collection of all bounded subsets of $X$, the topology $T$ on $X^*$ is generated by the family of seminorms of the form:
$$||phi||_A=sup_{xin A}|phi(x)|$$
where $phi$ is a continuous linear functional on $V$ and $A$ runs over the class $C$. Under what condition the dual $X^*$ is metrizable?
general-topology topological-vector-spaces
asked 2 days ago
Infinity
580313
put on hold as off-topic by user302797, Cesareo, Brahadeesh, Omran Kouba, Did yesterday
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – user302797, Cesareo, Brahadeesh, Omran Kouba, Did
If this question can be reworded to fit the rules in the help center, please edit the question.
1
For any locally convex and metrizable space $X$, its dual space is metrizable if and only if $X$ is normable. See also here.
– p4sch
2 days ago
What topology do you put on $X^ast$?
– Henno Brandsma
yesterday
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I edited my question including sufficient details.
Let $X$ be a metrizable topological vector space and $X^*$ be its dual space. Let $C$ be a collection of all bounded subsets of $X$, the topology $T$ on $X^*$ is generated by the family of seminorms of the form:
$$||phi||_A=sup_{xin A}|phi(x)|$$
where $phi$ is a continuous linear functional on $V$ and $A$ runs over the class $C$. Under what condition the dual $X^*$ is metrizable?
general-topology topological-vector-spaces
asked 2 days ago
Infinity
580313
I edited my question including sufficient details.
Let $X$ be a metrizable topological vector space and $X^*$ be its dual space. Let $C$ be a collection of all bounded subsets of $X$, the topology $T$ on $X^*$ is generated by the family of seminorms of the form:
$$||phi||_A=sup_{xin A}|phi(x)|$$
where $phi$ is a continuous linear functional on $V$ and $A$ runs over the class $C$. Under what condition the dual $X^*$ is metrizable?
general-topology topological-vector-spaces
general-topology topological-vector-spaces
asked 2 days ago
Infinity
580313
asked 2 days ago
Infinity
580313
edited 18 hours ago
asked 2 days ago
Infinity
580313
asked 2 days ago
Infinity
580313
asked 2 days ago
Infinity
580313
580313
put on hold as off-topic by user302797, Cesareo, Brahadeesh, Omran Kouba, Did yesterday
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – user302797, Cesareo, Brahadeesh, Omran Kouba, Did
If this question can be reworded to fit the rules in the help center, please edit the question.
put on hold as off-topic by user302797, Cesareo, Brahadeesh, Omran Kouba, Did yesterday
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – user302797, Cesareo, Brahadeesh, Omran Kouba, Did
If this question can be reworded to fit the rules in the help center, please edit the question.
1
For any locally convex and metrizable space $X$, its dual space is metrizable if and only if $X$ is normable. See also here.
– p4sch
2 days ago
What topology do you put on $X^ast$?
– Henno Brandsma
yesterday
add a comment |
1
For any locally convex and metrizable space $X$, its dual space is metrizable if and only if $X$ is normable. See also here.
– p4sch
2 days ago
What topology do you put on $X^ast$?
– Henno Brandsma
yesterday
1
1
For any locally convex and metrizable space $X$, its dual space is metrizable if and only if $X$ is normable. See also here.
– p4sch
2 days ago
For any locally convex and metrizable space $X$, its dual space is metrizable if and only if $X$ is normable. See also here.
– p4sch
2 days ago
What topology do you put on $X^ast$?
– Henno Brandsma
yesterday
What topology do you put on $X^ast$?
– Henno Brandsma
yesterday
add a comment |
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For any locally convex and metrizable space $X$, its dual space is metrizable if and only if $X$ is normable. See also here.
– p4sch
2 days ago
What topology do you put on $X^ast$?
– Henno Brandsma
yesterday