Are equivalent metrics on a Fréchet space strongly equivalent?
Suppose $(p_n)_n$ and $(q_n)$ are two increasing sequences of seminorms on a Fréchet space $X$. Each sequence can be used to define a (complete) metric on $X$. For example,
$$d_p(x,y)=sum_{n=1}^infty frac{1}{2^n} min(1,p_n(x-y))$$
and
$$d_q(x,y)=sum_{n=1}^infty frac{1}{2^n} min(1,q_n(x-y))$$
Must these two metrics be strongly equivalent, meaning there exist positive numbers $a$ and $b$ such that
$$a d_p(x,y) leq d_q(x,y) leq b d_p(x,y) ?$$
general-topology functional-analysis metric-spaces
edited Jan 5 at 23:47
Bernard
118k639112
asked Jan 5 at 23:34
user122916user122916
431314
add a comment |
Suppose $(p_n)_n$ and $(q_n)$ are two increasing sequences of seminorms on a Fréchet space $X$. Each sequence can be used to define a (complete) metric on $X$. For example,
$$d_p(x,y)=sum_{n=1}^infty frac{1}{2^n} min(1,p_n(x-y))$$
and
$$d_q(x,y)=sum_{n=1}^infty frac{1}{2^n} min(1,q_n(x-y))$$
Must these two metrics be strongly equivalent, meaning there exist positive numbers $a$ and $b$ such that
$$a d_p(x,y) leq d_q(x,y) leq b d_p(x,y) ?$$
general-topology functional-analysis metric-spaces
edited Jan 5 at 23:47
Bernard
118k639112
asked Jan 5 at 23:34
user122916user122916
431314
1
The question that you can ask is whether a continues linear map between Frechet spaces is necesssarily bounded, here the linear map you would consider is the identity map.
– pitariver
2 days ago
add a comment |
Suppose $(p_n)_n$ and $(q_n)$ are two increasing sequences of seminorms on a Fréchet space $X$. Each sequence can be used to define a (complete) metric on $X$. For example,
$$d_p(x,y)=sum_{n=1}^infty frac{1}{2^n} min(1,p_n(x-y))$$
and
$$d_q(x,y)=sum_{n=1}^infty frac{1}{2^n} min(1,q_n(x-y))$$
Must these two metrics be strongly equivalent, meaning there exist positive numbers $a$ and $b$ such that
$$a d_p(x,y) leq d_q(x,y) leq b d_p(x,y) ?$$
general-topology functional-analysis metric-spaces
edited Jan 5 at 23:47
Bernard
118k639112
asked Jan 5 at 23:34
user122916user122916
431314
Suppose $(p_n)_n$ and $(q_n)$ are two increasing sequences of seminorms on a Fréchet space $X$. Each sequence can be used to define a (complete) metric on $X$. For example,
$$d_p(x,y)=sum_{n=1}^infty frac{1}{2^n} min(1,p_n(x-y))$$
and
$$d_q(x,y)=sum_{n=1}^infty frac{1}{2^n} min(1,q_n(x-y))$$
Must these two metrics be strongly equivalent, meaning there exist positive numbers $a$ and $b$ such that
$$a d_p(x,y) leq d_q(x,y) leq b d_p(x,y) ?$$
general-topology functional-analysis metric-spaces
general-topology functional-analysis metric-spaces
edited Jan 5 at 23:47
Bernard
118k639112
asked Jan 5 at 23:34
user122916user122916
431314
edited Jan 5 at 23:47
Bernard
118k639112
asked Jan 5 at 23:34
user122916user122916
431314
edited Jan 5 at 23:47
Bernard
118k639112
edited Jan 5 at 23:47
Bernard
118k639112
edited Jan 5 at 23:47
Bernard
118k639112
118k639112
asked Jan 5 at 23:34
user122916user122916
431314
asked Jan 5 at 23:34
user122916user122916
431314
asked Jan 5 at 23:34
user122916user122916
431314
431314
1
The question that you can ask is whether a continues linear map between Frechet spaces is necesssarily bounded, here the linear map you would consider is the identity map.
– pitariver
2 days ago
add a comment |
1
The question that you can ask is whether a continues linear map between Frechet spaces is necesssarily bounded, here the linear map you would consider is the identity map.
– pitariver
2 days ago
1
1
The question that you can ask is whether a continues linear map between Frechet spaces is necesssarily bounded, here the linear map you would consider is the identity map.
– pitariver
2 days ago
The question that you can ask is whether a continues linear map between Frechet spaces is necesssarily bounded, here the linear map you would consider is the identity map.
– pitariver
2 days ago
add a comment |
1 Answer
1
active
oldest
votes
No. There are infinite-dimensional vector spaces $V$ carrying two non-equivalent norms $p$ and $q$ such that both $(V,p)$ and $(V,q)$ are Banach spaces. See
https://mathoverflow.net/q/184464.
Let $p_n = p, q_n = q$ for all $n$. Then $d_p(x,y) = min(1,p(x - y)), d_q(x,y) = min(1,q(x - y))$. These are not strongly equivalent.
answered 2 days ago
Paul FrostPaul Frost
9,6852732
I think the intended question (or the one I infer from the title) is whether two metrics that define Frechet spaces that have the exact same topology (same open sets) are necessarily strongly equivalent? (this is true for normed spaces) Do you have an answer or I should ask it as a separate questions?
– pitariver
2 days ago
@pitariver You are right, the title does not agree with the text of the question. So you should ask a new question.
– Paul Frost
2 days ago
add a comment |
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1 Answer
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1 Answer
1
active
oldest
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active
oldest
votes
No. There are infinite-dimensional vector spaces $V$ carrying two non-equivalent norms $p$ and $q$ such that both $(V,p)$ and $(V,q)$ are Banach spaces. See
https://mathoverflow.net/q/184464.
Let $p_n = p, q_n = q$ for all $n$. Then $d_p(x,y) = min(1,p(x - y)), d_q(x,y) = min(1,q(x - y))$. These are not strongly equivalent.
answered 2 days ago
Paul FrostPaul Frost
9,6852732
I think the intended question (or the one I infer from the title) is whether two metrics that define Frechet spaces that have the exact same topology (same open sets) are necessarily strongly equivalent? (this is true for normed spaces) Do you have an answer or I should ask it as a separate questions?
– pitariver
2 days ago
@pitariver You are right, the title does not agree with the text of the question. So you should ask a new question.
– Paul Frost
2 days ago
add a comment |
No. There are infinite-dimensional vector spaces $V$ carrying two non-equivalent norms $p$ and $q$ such that both $(V,p)$ and $(V,q)$ are Banach spaces. See
https://mathoverflow.net/q/184464.
Let $p_n = p, q_n = q$ for all $n$. Then $d_p(x,y) = min(1,p(x - y)), d_q(x,y) = min(1,q(x - y))$. These are not strongly equivalent.
answered 2 days ago
Paul FrostPaul Frost
9,6852732
I think the intended question (or the one I infer from the title) is whether two metrics that define Frechet spaces that have the exact same topology (same open sets) are necessarily strongly equivalent? (this is true for normed spaces) Do you have an answer or I should ask it as a separate questions?
– pitariver
2 days ago
@pitariver You are right, the title does not agree with the text of the question. So you should ask a new question.
– Paul Frost
2 days ago
add a comment |
No. There are infinite-dimensional vector spaces $V$ carrying two non-equivalent norms $p$ and $q$ such that both $(V,p)$ and $(V,q)$ are Banach spaces. See
https://mathoverflow.net/q/184464.
Let $p_n = p, q_n = q$ for all $n$. Then $d_p(x,y) = min(1,p(x - y)), d_q(x,y) = min(1,q(x - y))$. These are not strongly equivalent.
answered 2 days ago
Paul FrostPaul Frost
9,6852732
No. There are infinite-dimensional vector spaces $V$ carrying two non-equivalent norms $p$ and $q$ such that both $(V,p)$ and $(V,q)$ are Banach spaces. See
https://mathoverflow.net/q/184464.
Let $p_n = p, q_n = q$ for all $n$. Then $d_p(x,y) = min(1,p(x - y)), d_q(x,y) = min(1,q(x - y))$. These are not strongly equivalent.
answered 2 days ago
Paul FrostPaul Frost
9,6852732
answered 2 days ago
Paul FrostPaul Frost
9,6852732
answered 2 days ago
Paul FrostPaul Frost
9,6852732
answered 2 days ago
Paul FrostPaul Frost
9,6852732
9,6852732
I think the intended question (or the one I infer from the title) is whether two metrics that define Frechet spaces that have the exact same topology (same open sets) are necessarily strongly equivalent? (this is true for normed spaces) Do you have an answer or I should ask it as a separate questions?
– pitariver
2 days ago
@pitariver You are right, the title does not agree with the text of the question. So you should ask a new question.
– Paul Frost
2 days ago
add a comment |
I think the intended question (or the one I infer from the title) is whether two metrics that define Frechet spaces that have the exact same topology (same open sets) are necessarily strongly equivalent? (this is true for normed spaces) Do you have an answer or I should ask it as a separate questions?
– pitariver
2 days ago
@pitariver You are right, the title does not agree with the text of the question. So you should ask a new question.
– Paul Frost
2 days ago
I think the intended question (or the one I infer from the title) is whether two metrics that define Frechet spaces that have the exact same topology (same open sets) are necessarily strongly equivalent? (this is true for normed spaces) Do you have an answer or I should ask it as a separate questions?
– pitariver
2 days ago
I think the intended question (or the one I infer from the title) is whether two metrics that define Frechet spaces that have the exact same topology (same open sets) are necessarily strongly equivalent? (this is true for normed spaces) Do you have an answer or I should ask it as a separate questions?
– pitariver
2 days ago
@pitariver You are right, the title does not agree with the text of the question. So you should ask a new question.
– Paul Frost
2 days ago
@pitariver You are right, the title does not agree with the text of the question. So you should ask a new question.
– Paul Frost
2 days ago
add a comment |
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1
The question that you can ask is whether a continues linear map between Frechet spaces is necesssarily bounded, here the linear map you would consider is the identity map.
– pitariver
2 days ago