Are there any vector spaces that cannot be given a norm that makes the vector space a complete metric space?
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And if so, how can one prove that there is no such norm? I suppose one can use the form of Baire Category Theorem which states that a complete metric space cannot be written as a countable union of nowhere dense subsets to show that the metric space defined by the given vector space and norm is not complete but I have no idea on how to generalise this idea to all possible norms.
general-topology metric-spaces normed-spaces baire-category
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add a comment |
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And if so, how can one prove that there is no such norm? I suppose one can use the form of Baire Category Theorem which states that a complete metric space cannot be written as a countable union of nowhere dense subsets to show that the metric space defined by the given vector space and norm is not complete but I have no idea on how to generalise this idea to all possible norms.
general-topology metric-spaces normed-spaces baire-category
$endgroup$
add a comment |
$begingroup$
And if so, how can one prove that there is no such norm? I suppose one can use the form of Baire Category Theorem which states that a complete metric space cannot be written as a countable union of nowhere dense subsets to show that the metric space defined by the given vector space and norm is not complete but I have no idea on how to generalise this idea to all possible norms.
general-topology metric-spaces normed-spaces baire-category
$endgroup$
And if so, how can one prove that there is no such norm? I suppose one can use the form of Baire Category Theorem which states that a complete metric space cannot be written as a countable union of nowhere dense subsets to show that the metric space defined by the given vector space and norm is not complete but I have no idea on how to generalise this idea to all possible norms.
general-topology metric-spaces normed-spaces baire-category
general-topology metric-spaces normed-spaces baire-category
asked Dec 19 '18 at 18:46
2chromatic2chromatic
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3 Answers
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If a space admits an enumerable non finite basis $(e_i)_{iin bf N}$ it cannot be complete, as it is the enuramble union of finite dimensional vector spaces $V_n= vect (e_1,...e_n)$.
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A "natural" example is $mathbb{K}[X]$, $mathbb{K=R,C}$
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– Max
Dec 19 '18 at 20:50
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Yes. Take the space $c_{00}$ of all sequences $(a_n)_{ninmathbb N}$ of real numbers such that $a_n=0$ if $ngg1$. For each $ninmathbb N$, let $e(n)$ be the sequence such that its $n$th is $1$, where as all other terms are equal to $0$. Then, for any norm $lVertcdotrVert$ on $c_{00}$, the series$$sum_{n=0}^inftyfrac{e(n)}{n^2bigllVert e(n)bigrrVert}$$is a Cauchy series of elements of $c_{00}$ which doesn't converge in $c_{00}$.
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The vector space $sigma$ of all sequences in $mathbb{R}^omega$ with the property that ${n: x_n neq 0}$ is finite (usually taken in the subspace topology induced from the product (the $sigma$-product this is sometimes called)) is a standard example.
It's a countable union of finite-dimensional subspaces. In any norm topology on $sigma$ the finite-dimensional subspaces are closed and have empty interior, so in any such norm the space cannot be Baire (and so cannot be complete).
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3 Answers
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3 Answers
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active
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active
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active
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$begingroup$
If a space admits an enumerable non finite basis $(e_i)_{iin bf N}$ it cannot be complete, as it is the enuramble union of finite dimensional vector spaces $V_n= vect (e_1,...e_n)$.
$endgroup$
$begingroup$
A "natural" example is $mathbb{K}[X]$, $mathbb{K=R,C}$
$endgroup$
– Max
Dec 19 '18 at 20:50
add a comment |
$begingroup$
If a space admits an enumerable non finite basis $(e_i)_{iin bf N}$ it cannot be complete, as it is the enuramble union of finite dimensional vector spaces $V_n= vect (e_1,...e_n)$.
$endgroup$
$begingroup$
A "natural" example is $mathbb{K}[X]$, $mathbb{K=R,C}$
$endgroup$
– Max
Dec 19 '18 at 20:50
add a comment |
$begingroup$
If a space admits an enumerable non finite basis $(e_i)_{iin bf N}$ it cannot be complete, as it is the enuramble union of finite dimensional vector spaces $V_n= vect (e_1,...e_n)$.
$endgroup$
If a space admits an enumerable non finite basis $(e_i)_{iin bf N}$ it cannot be complete, as it is the enuramble union of finite dimensional vector spaces $V_n= vect (e_1,...e_n)$.
answered Dec 19 '18 at 18:50
ThomasThomas
4,102510
4,102510
$begingroup$
A "natural" example is $mathbb{K}[X]$, $mathbb{K=R,C}$
$endgroup$
– Max
Dec 19 '18 at 20:50
add a comment |
$begingroup$
A "natural" example is $mathbb{K}[X]$, $mathbb{K=R,C}$
$endgroup$
– Max
Dec 19 '18 at 20:50
$begingroup$
A "natural" example is $mathbb{K}[X]$, $mathbb{K=R,C}$
$endgroup$
– Max
Dec 19 '18 at 20:50
$begingroup$
A "natural" example is $mathbb{K}[X]$, $mathbb{K=R,C}$
$endgroup$
– Max
Dec 19 '18 at 20:50
add a comment |
$begingroup$
Yes. Take the space $c_{00}$ of all sequences $(a_n)_{ninmathbb N}$ of real numbers such that $a_n=0$ if $ngg1$. For each $ninmathbb N$, let $e(n)$ be the sequence such that its $n$th is $1$, where as all other terms are equal to $0$. Then, for any norm $lVertcdotrVert$ on $c_{00}$, the series$$sum_{n=0}^inftyfrac{e(n)}{n^2bigllVert e(n)bigrrVert}$$is a Cauchy series of elements of $c_{00}$ which doesn't converge in $c_{00}$.
$endgroup$
add a comment |
$begingroup$
Yes. Take the space $c_{00}$ of all sequences $(a_n)_{ninmathbb N}$ of real numbers such that $a_n=0$ if $ngg1$. For each $ninmathbb N$, let $e(n)$ be the sequence such that its $n$th is $1$, where as all other terms are equal to $0$. Then, for any norm $lVertcdotrVert$ on $c_{00}$, the series$$sum_{n=0}^inftyfrac{e(n)}{n^2bigllVert e(n)bigrrVert}$$is a Cauchy series of elements of $c_{00}$ which doesn't converge in $c_{00}$.
$endgroup$
add a comment |
$begingroup$
Yes. Take the space $c_{00}$ of all sequences $(a_n)_{ninmathbb N}$ of real numbers such that $a_n=0$ if $ngg1$. For each $ninmathbb N$, let $e(n)$ be the sequence such that its $n$th is $1$, where as all other terms are equal to $0$. Then, for any norm $lVertcdotrVert$ on $c_{00}$, the series$$sum_{n=0}^inftyfrac{e(n)}{n^2bigllVert e(n)bigrrVert}$$is a Cauchy series of elements of $c_{00}$ which doesn't converge in $c_{00}$.
$endgroup$
Yes. Take the space $c_{00}$ of all sequences $(a_n)_{ninmathbb N}$ of real numbers such that $a_n=0$ if $ngg1$. For each $ninmathbb N$, let $e(n)$ be the sequence such that its $n$th is $1$, where as all other terms are equal to $0$. Then, for any norm $lVertcdotrVert$ on $c_{00}$, the series$$sum_{n=0}^inftyfrac{e(n)}{n^2bigllVert e(n)bigrrVert}$$is a Cauchy series of elements of $c_{00}$ which doesn't converge in $c_{00}$.
answered Dec 19 '18 at 18:57
José Carlos SantosJosé Carlos Santos
164k22132235
164k22132235
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$begingroup$
The vector space $sigma$ of all sequences in $mathbb{R}^omega$ with the property that ${n: x_n neq 0}$ is finite (usually taken in the subspace topology induced from the product (the $sigma$-product this is sometimes called)) is a standard example.
It's a countable union of finite-dimensional subspaces. In any norm topology on $sigma$ the finite-dimensional subspaces are closed and have empty interior, so in any such norm the space cannot be Baire (and so cannot be complete).
$endgroup$
add a comment |
$begingroup$
The vector space $sigma$ of all sequences in $mathbb{R}^omega$ with the property that ${n: x_n neq 0}$ is finite (usually taken in the subspace topology induced from the product (the $sigma$-product this is sometimes called)) is a standard example.
It's a countable union of finite-dimensional subspaces. In any norm topology on $sigma$ the finite-dimensional subspaces are closed and have empty interior, so in any such norm the space cannot be Baire (and so cannot be complete).
$endgroup$
add a comment |
$begingroup$
The vector space $sigma$ of all sequences in $mathbb{R}^omega$ with the property that ${n: x_n neq 0}$ is finite (usually taken in the subspace topology induced from the product (the $sigma$-product this is sometimes called)) is a standard example.
It's a countable union of finite-dimensional subspaces. In any norm topology on $sigma$ the finite-dimensional subspaces are closed and have empty interior, so in any such norm the space cannot be Baire (and so cannot be complete).
$endgroup$
The vector space $sigma$ of all sequences in $mathbb{R}^omega$ with the property that ${n: x_n neq 0}$ is finite (usually taken in the subspace topology induced from the product (the $sigma$-product this is sometimes called)) is a standard example.
It's a countable union of finite-dimensional subspaces. In any norm topology on $sigma$ the finite-dimensional subspaces are closed and have empty interior, so in any such norm the space cannot be Baire (and so cannot be complete).
answered Dec 19 '18 at 22:05
Henno BrandsmaHenno Brandsma
111k348118
111k348118
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