Are equivalent metrics on a Fréchet space strongly equivalent?
$begingroup$
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
124k741117
asked Jan 5 at 23:34
user122916user122916
431315
$endgroup$
add a comment |
$begingroup$
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
124k741117
asked Jan 5 at 23:34
user122916user122916
431315
$endgroup$
1
$begingroup$
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.
$endgroup$
– pitariver
Jan 6 at 7:36
add a comment |
$begingroup$
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
124k741117
asked Jan 5 at 23:34
user122916user122916
431315
$endgroup$
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
124k741117
asked Jan 5 at 23:34
user122916user122916
431315
edited Jan 5 at 23:47
Bernard
124k741117
asked Jan 5 at 23:34
user122916user122916
431315
edited Jan 5 at 23:47
Bernard
124k741117
edited Jan 5 at 23:47
Bernard
124k741117
edited Jan 5 at 23:47
Bernard
124k741117
124k741117
asked Jan 5 at 23:34
user122916user122916
431315
asked Jan 5 at 23:34
user122916user122916
431315
asked Jan 5 at 23:34
user122916user122916
431315
431315
1
$begingroup$
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.
$endgroup$
– pitariver
Jan 6 at 7:36
add a comment |
1
$begingroup$
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.
$endgroup$
– pitariver
Jan 6 at 7:36
1
1
$begingroup$
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.
$endgroup$
– pitariver
Jan 6 at 7:36
$begingroup$
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.
$endgroup$
– pitariver
Jan 6 at 7:36
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
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 Jan 6 at 9:45
Paul FrostPaul Frost
12.5k31035
$endgroup$
$begingroup$
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?
$endgroup$
– pitariver
Jan 6 at 18:35
$begingroup$
@pitariver You are right, the title does not agree with the text of the question. So you should ask a new question.
$endgroup$
– Paul Frost
Jan 6 at 23:01
add a comment |
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1 Answer
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1 Answer
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$begingroup$
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 Jan 6 at 9:45
Paul FrostPaul Frost
12.5k31035
$endgroup$
$begingroup$
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?
$endgroup$
– pitariver
Jan 6 at 18:35
$begingroup$
@pitariver You are right, the title does not agree with the text of the question. So you should ask a new question.
$endgroup$
– Paul Frost
Jan 6 at 23:01
add a comment |
$begingroup$
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 Jan 6 at 9:45
Paul FrostPaul Frost
12.5k31035
$endgroup$
$begingroup$
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?
$endgroup$
– pitariver
Jan 6 at 18:35
$begingroup$
@pitariver You are right, the title does not agree with the text of the question. So you should ask a new question.
$endgroup$
– Paul Frost
Jan 6 at 23:01
add a comment |
$begingroup$
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 Jan 6 at 9:45
Paul FrostPaul Frost
12.5k31035
$endgroup$
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 Jan 6 at 9:45
Paul FrostPaul Frost
12.5k31035
answered Jan 6 at 9:45
Paul FrostPaul Frost
12.5k31035
answered Jan 6 at 9:45
Paul FrostPaul Frost
12.5k31035
answered Jan 6 at 9:45
Paul FrostPaul Frost
12.5k31035
12.5k31035
$begingroup$
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?
$endgroup$
– pitariver
Jan 6 at 18:35
$begingroup$
@pitariver You are right, the title does not agree with the text of the question. So you should ask a new question.
$endgroup$
– Paul Frost
Jan 6 at 23:01
add a comment |
$begingroup$
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?
$endgroup$
– pitariver
Jan 6 at 18:35
$begingroup$
@pitariver You are right, the title does not agree with the text of the question. So you should ask a new question.
$endgroup$
– Paul Frost
Jan 6 at 23:01
$begingroup$
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?
$endgroup$
– pitariver
Jan 6 at 18:35
$begingroup$
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?
$endgroup$
– pitariver
Jan 6 at 18:35
$begingroup$
@pitariver You are right, the title does not agree with the text of the question. So you should ask a new question.
$endgroup$
– Paul Frost
Jan 6 at 23:01
$begingroup$
@pitariver You are right, the title does not agree with the text of the question. So you should ask a new question.
$endgroup$
– Paul Frost
Jan 6 at 23:01
add a comment |
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$begingroup$
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.
$endgroup$
– pitariver
Jan 6 at 7:36