Equivalent criteria for being dense in $L_p(X)$
2
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Is there any equivalent criteria to show that subset $A$ in the Lebesgue space $L_p(X)$ is dense? In particular, I am interested in $L_2((0,1))$.
real-analysis functional-analysis hilbert-spaces lebesgue-integral banach-spaces
asked Jan 2 at 16:40
ershersh
423113
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add a comment |
2
$begingroup$
Is there any equivalent criteria to show that subset $A$ in the Lebesgue space $L_p(X)$ is dense? In particular, I am interested in $L_2((0,1))$.
real-analysis functional-analysis hilbert-spaces lebesgue-integral banach-spaces
asked Jan 2 at 16:40
ershersh
423113
$endgroup$
1
$begingroup$
For $X=L_2(0,1)$ and a set $A$ within, consider the orthogonal of A, namely $A^{perp}$, and the Hilbert space structure of X. $A$ is dense in $X$ iff $A^{perp}={0}$.
$endgroup$
– Marko Karbevski
Jan 2 at 16:44
$begingroup$
@MarkoKarbevski Where can I find the proof or how can I prove it?
$endgroup$
– ersh
Jan 2 at 19:48
1
$begingroup$
I forgot to mention one thing, if $A$ is dense then its orthogonal is trivial for any set $A$. However if $A^perp={0}$ we need an additional criteria in order to conclude: we want $A$ to be a vector subspace (otherwise choose the unit sphere, its orthogonal is trivial but it is not dense).
$endgroup$
– Marko Karbevski
Jan 2 at 20:45
1
$begingroup$
For the first part, let $A$ be dense, $a_0 in A^perp$. There exists a sequence $(a_n)$ of elements in $A$ such that $a_n to a$. 1) What is the value of $langle a_n,a rangle$ ? ; 2) What's its limit in terms of $|a| $? ; 3) Conclude. You should find the proofs here: math.stackexchange.com/questions/1315321/…
$endgroup$
– Marko Karbevski
Jan 2 at 20:45
$begingroup$
Thanks, I just needed this part "$A$ is dense then its orthogonal is trivial for any set $A$".
$endgroup$
– ersh
Jan 2 at 21:32
add a comment |
2
2
2
$begingroup$
Is there any equivalent criteria to show that subset $A$ in the Lebesgue space $L_p(X)$ is dense? In particular, I am interested in $L_2((0,1))$.
real-analysis functional-analysis hilbert-spaces lebesgue-integral banach-spaces
asked Jan 2 at 16:40
ershersh
423113
$endgroup$
Is there any equivalent criteria to show that subset $A$ in the Lebesgue space $L_p(X)$ is dense? In particular, I am interested in $L_2((0,1))$.
real-analysis functional-analysis hilbert-spaces lebesgue-integral banach-spaces
real-analysis functional-analysis hilbert-spaces lebesgue-integral banach-spaces
asked Jan 2 at 16:40
ershersh
423113
asked Jan 2 at 16:40
ershersh
423113
asked Jan 2 at 16:40
ershersh
423113
asked Jan 2 at 16:40
ershersh
423113
asked Jan 2 at 16:40
ershersh
423113
423113
1
$begingroup$
For $X=L_2(0,1)$ and a set $A$ within, consider the orthogonal of A, namely $A^{perp}$, and the Hilbert space structure of X. $A$ is dense in $X$ iff $A^{perp}={0}$.
$endgroup$
– Marko Karbevski
Jan 2 at 16:44
$begingroup$
@MarkoKarbevski Where can I find the proof or how can I prove it?
$endgroup$
– ersh
Jan 2 at 19:48
1
$begingroup$
I forgot to mention one thing, if $A$ is dense then its orthogonal is trivial for any set $A$. However if $A^perp={0}$ we need an additional criteria in order to conclude: we want $A$ to be a vector subspace (otherwise choose the unit sphere, its orthogonal is trivial but it is not dense).
$endgroup$
– Marko Karbevski
Jan 2 at 20:45
1
$begingroup$
For the first part, let $A$ be dense, $a_0 in A^perp$. There exists a sequence $(a_n)$ of elements in $A$ such that $a_n to a$. 1) What is the value of $langle a_n,a rangle$ ? ; 2) What's its limit in terms of $|a| $? ; 3) Conclude. You should find the proofs here: math.stackexchange.com/questions/1315321/…
$endgroup$
– Marko Karbevski
Jan 2 at 20:45
$begingroup$
Thanks, I just needed this part "$A$ is dense then its orthogonal is trivial for any set $A$".
$endgroup$
– ersh
Jan 2 at 21:32
add a comment |
1
$begingroup$
For $X=L_2(0,1)$ and a set $A$ within, consider the orthogonal of A, namely $A^{perp}$, and the Hilbert space structure of X. $A$ is dense in $X$ iff $A^{perp}={0}$.
$endgroup$
– Marko Karbevski
Jan 2 at 16:44
$begingroup$
@MarkoKarbevski Where can I find the proof or how can I prove it?
$endgroup$
– ersh
Jan 2 at 19:48
1
$begingroup$
I forgot to mention one thing, if $A$ is dense then its orthogonal is trivial for any set $A$. However if $A^perp={0}$ we need an additional criteria in order to conclude: we want $A$ to be a vector subspace (otherwise choose the unit sphere, its orthogonal is trivial but it is not dense).
$endgroup$
– Marko Karbevski
Jan 2 at 20:45
1
$begingroup$
For the first part, let $A$ be dense, $a_0 in A^perp$. There exists a sequence $(a_n)$ of elements in $A$ such that $a_n to a$. 1) What is the value of $langle a_n,a rangle$ ? ; 2) What's its limit in terms of $|a| $? ; 3) Conclude. You should find the proofs here: math.stackexchange.com/questions/1315321/…
$endgroup$
– Marko Karbevski
Jan 2 at 20:45
$begingroup$
Thanks, I just needed this part "$A$ is dense then its orthogonal is trivial for any set $A$".
$endgroup$
– ersh
Jan 2 at 21:32
1
1
$begingroup$
For $X=L_2(0,1)$ and a set $A$ within, consider the orthogonal of A, namely $A^{perp}$, and the Hilbert space structure of X. $A$ is dense in $X$ iff $A^{perp}={0}$.
$endgroup$
– Marko Karbevski
Jan 2 at 16:44
$begingroup$
For $X=L_2(0,1)$ and a set $A$ within, consider the orthogonal of A, namely $A^{perp}$, and the Hilbert space structure of X. $A$ is dense in $X$ iff $A^{perp}={0}$.
$endgroup$
– Marko Karbevski
Jan 2 at 16:44
$begingroup$
@MarkoKarbevski Where can I find the proof or how can I prove it?
$endgroup$
– ersh
Jan 2 at 19:48
$begingroup$
@MarkoKarbevski Where can I find the proof or how can I prove it?
$endgroup$
– ersh
Jan 2 at 19:48
1
1
$begingroup$
I forgot to mention one thing, if $A$ is dense then its orthogonal is trivial for any set $A$. However if $A^perp={0}$ we need an additional criteria in order to conclude: we want $A$ to be a vector subspace (otherwise choose the unit sphere, its orthogonal is trivial but it is not dense).
$endgroup$
– Marko Karbevski
Jan 2 at 20:45
$begingroup$
I forgot to mention one thing, if $A$ is dense then its orthogonal is trivial for any set $A$. However if $A^perp={0}$ we need an additional criteria in order to conclude: we want $A$ to be a vector subspace (otherwise choose the unit sphere, its orthogonal is trivial but it is not dense).
$endgroup$
– Marko Karbevski
Jan 2 at 20:45
1
1
$begingroup$
For the first part, let $A$ be dense, $a_0 in A^perp$. There exists a sequence $(a_n)$ of elements in $A$ such that $a_n to a$. 1) What is the value of $langle a_n,a rangle$ ? ; 2) What's its limit in terms of $|a| $? ; 3) Conclude. You should find the proofs here: math.stackexchange.com/questions/1315321/…
$endgroup$
– Marko Karbevski
Jan 2 at 20:45
$begingroup$
For the first part, let $A$ be dense, $a_0 in A^perp$. There exists a sequence $(a_n)$ of elements in $A$ such that $a_n to a$. 1) What is the value of $langle a_n,a rangle$ ? ; 2) What's its limit in terms of $|a| $? ; 3) Conclude. You should find the proofs here: math.stackexchange.com/questions/1315321/…
$endgroup$
– Marko Karbevski
Jan 2 at 20:45
$begingroup$
Thanks, I just needed this part "$A$ is dense then its orthogonal is trivial for any set $A$".
$endgroup$
– ersh
Jan 2 at 21:32
$begingroup$
Thanks, I just needed this part "$A$ is dense then its orthogonal is trivial for any set $A$".
$endgroup$
– ersh
Jan 2 at 21:32
add a comment |
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1
$begingroup$
For $X=L_2(0,1)$ and a set $A$ within, consider the orthogonal of A, namely $A^{perp}$, and the Hilbert space structure of X. $A$ is dense in $X$ iff $A^{perp}={0}$.
$endgroup$
– Marko Karbevski
Jan 2 at 16:44
$begingroup$
@MarkoKarbevski Where can I find the proof or how can I prove it?
$endgroup$
– ersh
Jan 2 at 19:48
1
$begingroup$
I forgot to mention one thing, if $A$ is dense then its orthogonal is trivial for any set $A$. However if $A^perp={0}$ we need an additional criteria in order to conclude: we want $A$ to be a vector subspace (otherwise choose the unit sphere, its orthogonal is trivial but it is not dense).
$endgroup$
– Marko Karbevski
Jan 2 at 20:45
1
$begingroup$
For the first part, let $A$ be dense, $a_0 in A^perp$. There exists a sequence $(a_n)$ of elements in $A$ such that $a_n to a$. 1) What is the value of $langle a_n,a rangle$ ? ; 2) What's its limit in terms of $|a| $? ; 3) Conclude. You should find the proofs here: math.stackexchange.com/questions/1315321/…
$endgroup$
– Marko Karbevski
Jan 2 at 20:45
$begingroup$
Thanks, I just needed this part "$A$ is dense then its orthogonal is trivial for any set $A$".
$endgroup$
– ersh
Jan 2 at 21:32