Any example of equivalent norms in infinite-dimensional vector space?
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I have read in this paper that the number of inequivalent norms in infinit-dimensional space is exactly $2^{dim X}$, In my guess if am true this mean that there are Equivalent norms in infinit dimensional space which I want to know one example of it ? and if there is no Equivalents norms in infinit dimensional vector space just any proof for that ?
vector-spaces normed-spaces
edited Dec 25 '18 at 16:26
Bernard
124k741117
asked Dec 25 '18 at 16:14
zeraoulia rafikzeraoulia rafik
2,37711133
$endgroup$
add a comment |
$begingroup$
I have read in this paper that the number of inequivalent norms in infinit-dimensional space is exactly $2^{dim X}$, In my guess if am true this mean that there are Equivalent norms in infinit dimensional space which I want to know one example of it ? and if there is no Equivalents norms in infinit dimensional vector space just any proof for that ?
vector-spaces normed-spaces
edited Dec 25 '18 at 16:26
Bernard
124k741117
asked Dec 25 '18 at 16:14
zeraoulia rafikzeraoulia rafik
2,37711133
$endgroup$
$begingroup$
Take a norm $|cdot |$ and $c>0$. The norm $c|cdot |$ is equivalent to $|cdot |$.
$endgroup$
– user608030
Dec 25 '18 at 16:19
add a comment |
$begingroup$
I have read in this paper that the number of inequivalent norms in infinit-dimensional space is exactly $2^{dim X}$, In my guess if am true this mean that there are Equivalent norms in infinit dimensional space which I want to know one example of it ? and if there is no Equivalents norms in infinit dimensional vector space just any proof for that ?
vector-spaces normed-spaces
edited Dec 25 '18 at 16:26
Bernard
124k741117
asked Dec 25 '18 at 16:14
zeraoulia rafikzeraoulia rafik
2,37711133
$endgroup$
I have read in this paper that the number of inequivalent norms in infinit-dimensional space is exactly $2^{dim X}$, In my guess if am true this mean that there are Equivalent norms in infinit dimensional space which I want to know one example of it ? and if there is no Equivalents norms in infinit dimensional vector space just any proof for that ?
vector-spaces normed-spaces
vector-spaces normed-spaces
edited Dec 25 '18 at 16:26
Bernard
124k741117
asked Dec 25 '18 at 16:14
zeraoulia rafikzeraoulia rafik
2,37711133
edited Dec 25 '18 at 16:26
Bernard
124k741117
asked Dec 25 '18 at 16:14
zeraoulia rafikzeraoulia rafik
2,37711133
edited Dec 25 '18 at 16:26
Bernard
124k741117
edited Dec 25 '18 at 16:26
Bernard
124k741117
edited Dec 25 '18 at 16:26
Bernard
124k741117
124k741117
asked Dec 25 '18 at 16:14
zeraoulia rafikzeraoulia rafik
2,37711133
asked Dec 25 '18 at 16:14
zeraoulia rafikzeraoulia rafik
2,37711133
asked Dec 25 '18 at 16:14
zeraoulia rafikzeraoulia rafik
2,37711133
2,37711133
$begingroup$
Take a norm $|cdot |$ and $c>0$. The norm $c|cdot |$ is equivalent to $|cdot |$.
$endgroup$
– user608030
Dec 25 '18 at 16:19
add a comment |
$begingroup$
Take a norm $|cdot |$ and $c>0$. The norm $c|cdot |$ is equivalent to $|cdot |$.
$endgroup$
– user608030
Dec 25 '18 at 16:19
$begingroup$
Take a norm $|cdot |$ and $c>0$. The norm $c|cdot |$ is equivalent to $|cdot |$.
$endgroup$
– user608030
Dec 25 '18 at 16:19
$begingroup$
Take a norm $|cdot |$ and $c>0$. The norm $c|cdot |$ is equivalent to $|cdot |$.
$endgroup$
– user608030
Dec 25 '18 at 16:19
add a comment |
2 Answers
2
active
oldest
votes
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If $lVertcdotrVert$ is any norm on any vector space $X$, then $lVertcdotrVert$ and $2lVertcdotrVert$ are equivalent norms.
answered Dec 25 '18 at 16:17
José Carlos SantosJosé Carlos Santos
173k23133241
$endgroup$
add a comment |
$begingroup$
Here's a silly example: define on $X=mathcal{C}^0([a,b],mathbb{R})$ the norm $lVertcdotrVert$ given by $sup_{xin [a,b]} f(x)=lVert frVert.$ Define a second norm $lVertcdotrVert'$ by $2sup_{xin [a,b]}f(x)=lVert frVert'$. Then indeed we have that there exist constants $c$ and $C$ such that
$$ clVert frVert'le lVert frVertle ClVert frVert'.$$
For instance, take $c=frac{1}{2}$ and $C=1$. More generally, if we have a normed space over $mathbb{R}$, a norm $lVertcdotrVert$ determines a family of equivalent norms $lambda lVertcdotrVert$ for $lambdain mathbb{R}$. This fact is independent of the dimensionality of the space.
answered Dec 25 '18 at 16:19
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.6k41741
$endgroup$
add a comment |
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2 Answers
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active
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2 Answers
2
active
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votes
$begingroup$
If $lVertcdotrVert$ is any norm on any vector space $X$, then $lVertcdotrVert$ and $2lVertcdotrVert$ are equivalent norms.
answered Dec 25 '18 at 16:17
José Carlos SantosJosé Carlos Santos
173k23133241
$endgroup$
add a comment |
$begingroup$
If $lVertcdotrVert$ is any norm on any vector space $X$, then $lVertcdotrVert$ and $2lVertcdotrVert$ are equivalent norms.
answered Dec 25 '18 at 16:17
José Carlos SantosJosé Carlos Santos
173k23133241
$endgroup$
add a comment |
$begingroup$
If $lVertcdotrVert$ is any norm on any vector space $X$, then $lVertcdotrVert$ and $2lVertcdotrVert$ are equivalent norms.
answered Dec 25 '18 at 16:17
José Carlos SantosJosé Carlos Santos
173k23133241
$endgroup$
If $lVertcdotrVert$ is any norm on any vector space $X$, then $lVertcdotrVert$ and $2lVertcdotrVert$ are equivalent norms.
answered Dec 25 '18 at 16:17
José Carlos SantosJosé Carlos Santos
173k23133241
answered Dec 25 '18 at 16:17
José Carlos SantosJosé Carlos Santos
173k23133241
answered Dec 25 '18 at 16:17
José Carlos SantosJosé Carlos Santos
173k23133241
answered Dec 25 '18 at 16:17
José Carlos SantosJosé Carlos Santos
173k23133241
173k23133241
add a comment |
add a comment |
$begingroup$
Here's a silly example: define on $X=mathcal{C}^0([a,b],mathbb{R})$ the norm $lVertcdotrVert$ given by $sup_{xin [a,b]} f(x)=lVert frVert.$ Define a second norm $lVertcdotrVert'$ by $2sup_{xin [a,b]}f(x)=lVert frVert'$. Then indeed we have that there exist constants $c$ and $C$ such that
$$ clVert frVert'le lVert frVertle ClVert frVert'.$$
For instance, take $c=frac{1}{2}$ and $C=1$. More generally, if we have a normed space over $mathbb{R}$, a norm $lVertcdotrVert$ determines a family of equivalent norms $lambda lVertcdotrVert$ for $lambdain mathbb{R}$. This fact is independent of the dimensionality of the space.
answered Dec 25 '18 at 16:19
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.6k41741
$endgroup$
add a comment |
$begingroup$
Here's a silly example: define on $X=mathcal{C}^0([a,b],mathbb{R})$ the norm $lVertcdotrVert$ given by $sup_{xin [a,b]} f(x)=lVert frVert.$ Define a second norm $lVertcdotrVert'$ by $2sup_{xin [a,b]}f(x)=lVert frVert'$. Then indeed we have that there exist constants $c$ and $C$ such that
$$ clVert frVert'le lVert frVertle ClVert frVert'.$$
For instance, take $c=frac{1}{2}$ and $C=1$. More generally, if we have a normed space over $mathbb{R}$, a norm $lVertcdotrVert$ determines a family of equivalent norms $lambda lVertcdotrVert$ for $lambdain mathbb{R}$. This fact is independent of the dimensionality of the space.
answered Dec 25 '18 at 16:19
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.6k41741
$endgroup$
add a comment |
$begingroup$
Here's a silly example: define on $X=mathcal{C}^0([a,b],mathbb{R})$ the norm $lVertcdotrVert$ given by $sup_{xin [a,b]} f(x)=lVert frVert.$ Define a second norm $lVertcdotrVert'$ by $2sup_{xin [a,b]}f(x)=lVert frVert'$. Then indeed we have that there exist constants $c$ and $C$ such that
$$ clVert frVert'le lVert frVertle ClVert frVert'.$$
For instance, take $c=frac{1}{2}$ and $C=1$. More generally, if we have a normed space over $mathbb{R}$, a norm $lVertcdotrVert$ determines a family of equivalent norms $lambda lVertcdotrVert$ for $lambdain mathbb{R}$. This fact is independent of the dimensionality of the space.
answered Dec 25 '18 at 16:19
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.6k41741
$endgroup$
Here's a silly example: define on $X=mathcal{C}^0([a,b],mathbb{R})$ the norm $lVertcdotrVert$ given by $sup_{xin [a,b]} f(x)=lVert frVert.$ Define a second norm $lVertcdotrVert'$ by $2sup_{xin [a,b]}f(x)=lVert frVert'$. Then indeed we have that there exist constants $c$ and $C$ such that
$$ clVert frVert'le lVert frVertle ClVert frVert'.$$
For instance, take $c=frac{1}{2}$ and $C=1$. More generally, if we have a normed space over $mathbb{R}$, a norm $lVertcdotrVert$ determines a family of equivalent norms $lambda lVertcdotrVert$ for $lambdain mathbb{R}$. This fact is independent of the dimensionality of the space.
answered Dec 25 '18 at 16:19
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.6k41741
answered Dec 25 '18 at 16:19
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.6k41741
answered Dec 25 '18 at 16:19
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.6k41741
answered Dec 25 '18 at 16:19
Antonios-Alexandros RobotisAntonios-Alexandros Robotis
10.6k41741
10.6k41741
add a comment |
add a comment |
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$begingroup$
Take a norm $|cdot |$ and $c>0$. The norm $c|cdot |$ is equivalent to $|cdot |$.
$endgroup$
– user608030
Dec 25 '18 at 16:19