Example of $U subseteq V$ such that $V$ is infinite-dimensional and $U^0 = V'$ but $U neq {0}$.
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I am working through Axler's Linear Algebra Done Right, where he uses the notation $U^0$ for the annihilator subspace of the dual space $V'$ such that if $varphi in U^0$, then $U subseteq text{null}(varphi)$. Note that it is true that if $V$ is finite-dimensional, then $U^0 = V'$ implies that $U = {0}$. However, I am curious to find an example when $V$ is not finite-dimensional where $U$ needs not be equal to ${0}$. Thank you in advance for your help!
PS - if there is a proof that $U = {0}$ if $U^0 = V'$ when $V$ is infinite dimensional, then I would be interested in that too!
linear-algebra duality-theorems dual-spaces
asked Dec 17 '18 at 11:21
tucsonman101tucsonman101
134
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add a comment |
$begingroup$
I am working through Axler's Linear Algebra Done Right, where he uses the notation $U^0$ for the annihilator subspace of the dual space $V'$ such that if $varphi in U^0$, then $U subseteq text{null}(varphi)$. Note that it is true that if $V$ is finite-dimensional, then $U^0 = V'$ implies that $U = {0}$. However, I am curious to find an example when $V$ is not finite-dimensional where $U$ needs not be equal to ${0}$. Thank you in advance for your help!
PS - if there is a proof that $U = {0}$ if $U^0 = V'$ when $V$ is infinite dimensional, then I would be interested in that too!
linear-algebra duality-theorems dual-spaces
asked Dec 17 '18 at 11:21
tucsonman101tucsonman101
134
$endgroup$
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Related : math.stackexchange.com/questions/142904/…
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– Arnaud D.
Dec 17 '18 at 11:29
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Thank you for the link - I might need to review it more, but could you provide a concrete example for my above question?
$endgroup$
– tucsonman101
Dec 17 '18 at 11:36
add a comment |
$begingroup$
I am working through Axler's Linear Algebra Done Right, where he uses the notation $U^0$ for the annihilator subspace of the dual space $V'$ such that if $varphi in U^0$, then $U subseteq text{null}(varphi)$. Note that it is true that if $V$ is finite-dimensional, then $U^0 = V'$ implies that $U = {0}$. However, I am curious to find an example when $V$ is not finite-dimensional where $U$ needs not be equal to ${0}$. Thank you in advance for your help!
PS - if there is a proof that $U = {0}$ if $U^0 = V'$ when $V$ is infinite dimensional, then I would be interested in that too!
linear-algebra duality-theorems dual-spaces
asked Dec 17 '18 at 11:21
tucsonman101tucsonman101
134
$endgroup$
I am working through Axler's Linear Algebra Done Right, where he uses the notation $U^0$ for the annihilator subspace of the dual space $V'$ such that if $varphi in U^0$, then $U subseteq text{null}(varphi)$. Note that it is true that if $V$ is finite-dimensional, then $U^0 = V'$ implies that $U = {0}$. However, I am curious to find an example when $V$ is not finite-dimensional where $U$ needs not be equal to ${0}$. Thank you in advance for your help!
PS - if there is a proof that $U = {0}$ if $U^0 = V'$ when $V$ is infinite dimensional, then I would be interested in that too!
linear-algebra duality-theorems dual-spaces
linear-algebra duality-theorems dual-spaces
asked Dec 17 '18 at 11:21
tucsonman101tucsonman101
134
asked Dec 17 '18 at 11:21
tucsonman101tucsonman101
134
asked Dec 17 '18 at 11:21
tucsonman101tucsonman101
134
asked Dec 17 '18 at 11:21
tucsonman101tucsonman101
134
asked Dec 17 '18 at 11:21
tucsonman101tucsonman101
134
134
$begingroup$
Related : math.stackexchange.com/questions/142904/…
$endgroup$
– Arnaud D.
Dec 17 '18 at 11:29
$begingroup$
Thank you for the link - I might need to review it more, but could you provide a concrete example for my above question?
$endgroup$
– tucsonman101
Dec 17 '18 at 11:36
add a comment |
$begingroup$
Related : math.stackexchange.com/questions/142904/…
$endgroup$
– Arnaud D.
Dec 17 '18 at 11:29
$begingroup$
Thank you for the link - I might need to review it more, but could you provide a concrete example for my above question?
$endgroup$
– tucsonman101
Dec 17 '18 at 11:36
$begingroup$
Related : math.stackexchange.com/questions/142904/…
$endgroup$
– Arnaud D.
Dec 17 '18 at 11:29
$begingroup$
Related : math.stackexchange.com/questions/142904/…
$endgroup$
– Arnaud D.
Dec 17 '18 at 11:29
$begingroup$
Thank you for the link - I might need to review it more, but could you provide a concrete example for my above question?
$endgroup$
– tucsonman101
Dec 17 '18 at 11:36
$begingroup$
Thank you for the link - I might need to review it more, but could you provide a concrete example for my above question?
$endgroup$
– tucsonman101
Dec 17 '18 at 11:36
add a comment |
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Related : math.stackexchange.com/questions/142904/…
$endgroup$
– Arnaud D.
Dec 17 '18 at 11:29
$begingroup$
Thank you for the link - I might need to review it more, but could you provide a concrete example for my above question?
$endgroup$
– tucsonman101
Dec 17 '18 at 11:36