Weak vs. strong convergence in the proof of the Hodge decomposition theorem in Warner, p.224
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I'm reading the proof of the Hodge decomposition theorem in Warner, Foundations of Differentiable Manifolds and Lie Groups. At p.224, it is shown that
$$
lim_{jtoinfty} langlebeta_j,psirangle = langlebeta,psirangle,qquadforallpsiin E^p(M)
$$
Then, Warner says :
Consequently, $beta_j to beta$
without specifying if this is a strong or a weak convergence.
It seems that it should be a weak convergence.
But then, the sentence after is :
Since $|beta_j|=1$ and $beta_jin (H^p)^perp$, it follows that $|beta|=1$ and $betain (H^p)^perp$.
which needs strong convergence. What am I missing?
A bit of context : we are on a smooth Riemannian oriented closed manifold. $E^p(M)$ denotes the space of differential $p$-forms on $M$. $H^p$ denotes the kernel of the Hodge-de Rham Laplacian $Delta=ddelta+delta d$. $(H^p)^perp$ denotes the $L^2$-perpendicular to $H^p$. $(beta_j)$ is a Cauchy sequence in $(H^p)^perp$.
For the whole picture, the best is to read the page of the book.
functional-analysis differential-geometry hodge-theory
asked Dec 3 '18 at 0:43
NACNAC
1,110412
$endgroup$
add a comment |
$begingroup$
I'm reading the proof of the Hodge decomposition theorem in Warner, Foundations of Differentiable Manifolds and Lie Groups. At p.224, it is shown that
$$
lim_{jtoinfty} langlebeta_j,psirangle = langlebeta,psirangle,qquadforallpsiin E^p(M)
$$
Then, Warner says :
Consequently, $beta_j to beta$
without specifying if this is a strong or a weak convergence.
It seems that it should be a weak convergence.
But then, the sentence after is :
Since $|beta_j|=1$ and $beta_jin (H^p)^perp$, it follows that $|beta|=1$ and $betain (H^p)^perp$.
which needs strong convergence. What am I missing?
A bit of context : we are on a smooth Riemannian oriented closed manifold. $E^p(M)$ denotes the space of differential $p$-forms on $M$. $H^p$ denotes the kernel of the Hodge-de Rham Laplacian $Delta=ddelta+delta d$. $(H^p)^perp$ denotes the $L^2$-perpendicular to $H^p$. $(beta_j)$ is a Cauchy sequence in $(H^p)^perp$.
For the whole picture, the best is to read the page of the book.
functional-analysis differential-geometry hodge-theory
asked Dec 3 '18 at 0:43
NACNAC
1,110412
$endgroup$
$begingroup$
What are $H^p, E^p$? smooth functions defined around $p$ or... ?
$endgroup$
– user25959
Dec 3 '18 at 1:44
$begingroup$
@user25959 I added a bit of context in the question.
$endgroup$
– NAC
Dec 3 '18 at 1:50
add a comment |
$begingroup$
I'm reading the proof of the Hodge decomposition theorem in Warner, Foundations of Differentiable Manifolds and Lie Groups. At p.224, it is shown that
$$
lim_{jtoinfty} langlebeta_j,psirangle = langlebeta,psirangle,qquadforallpsiin E^p(M)
$$
Then, Warner says :
Consequently, $beta_j to beta$
without specifying if this is a strong or a weak convergence.
It seems that it should be a weak convergence.
But then, the sentence after is :
Since $|beta_j|=1$ and $beta_jin (H^p)^perp$, it follows that $|beta|=1$ and $betain (H^p)^perp$.
which needs strong convergence. What am I missing?
A bit of context : we are on a smooth Riemannian oriented closed manifold. $E^p(M)$ denotes the space of differential $p$-forms on $M$. $H^p$ denotes the kernel of the Hodge-de Rham Laplacian $Delta=ddelta+delta d$. $(H^p)^perp$ denotes the $L^2$-perpendicular to $H^p$. $(beta_j)$ is a Cauchy sequence in $(H^p)^perp$.
For the whole picture, the best is to read the page of the book.
functional-analysis differential-geometry hodge-theory
asked Dec 3 '18 at 0:43
NACNAC
1,110412
$endgroup$
I'm reading the proof of the Hodge decomposition theorem in Warner, Foundations of Differentiable Manifolds and Lie Groups. At p.224, it is shown that
$$
lim_{jtoinfty} langlebeta_j,psirangle = langlebeta,psirangle,qquadforallpsiin E^p(M)
$$
Then, Warner says :
Consequently, $beta_j to beta$
without specifying if this is a strong or a weak convergence.
It seems that it should be a weak convergence.
But then, the sentence after is :
Since $|beta_j|=1$ and $beta_jin (H^p)^perp$, it follows that $|beta|=1$ and $betain (H^p)^perp$.
which needs strong convergence. What am I missing?
A bit of context : we are on a smooth Riemannian oriented closed manifold. $E^p(M)$ denotes the space of differential $p$-forms on $M$. $H^p$ denotes the kernel of the Hodge-de Rham Laplacian $Delta=ddelta+delta d$. $(H^p)^perp$ denotes the $L^2$-perpendicular to $H^p$. $(beta_j)$ is a Cauchy sequence in $(H^p)^perp$.
For the whole picture, the best is to read the page of the book.
functional-analysis differential-geometry hodge-theory
functional-analysis differential-geometry hodge-theory
asked Dec 3 '18 at 0:43
NACNAC
1,110412
asked Dec 3 '18 at 0:43
NACNAC
1,110412
edited Dec 3 '18 at 1:49
NAC
asked Dec 3 '18 at 0:43
NACNAC
1,110412
asked Dec 3 '18 at 0:43
NACNAC
1,110412
asked Dec 3 '18 at 0:43
NACNAC
1,110412
1,110412
$begingroup$
What are $H^p, E^p$? smooth functions defined around $p$ or... ?
$endgroup$
– user25959
Dec 3 '18 at 1:44
$begingroup$
@user25959 I added a bit of context in the question.
$endgroup$
– NAC
Dec 3 '18 at 1:50
add a comment |
$begingroup$
What are $H^p, E^p$? smooth functions defined around $p$ or... ?
$endgroup$
– user25959
Dec 3 '18 at 1:44
$begingroup$
@user25959 I added a bit of context in the question.
$endgroup$
– NAC
Dec 3 '18 at 1:50
$begingroup$
What are $H^p, E^p$? smooth functions defined around $p$ or... ?
$endgroup$
– user25959
Dec 3 '18 at 1:44
$begingroup$
What are $H^p, E^p$? smooth functions defined around $p$ or... ?
$endgroup$
– user25959
Dec 3 '18 at 1:44
$begingroup$
@user25959 I added a bit of context in the question.
$endgroup$
– NAC
Dec 3 '18 at 1:50
$begingroup$
@user25959 I added a bit of context in the question.
$endgroup$
– NAC
Dec 3 '18 at 1:50
add a comment |
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$begingroup$
What are $H^p, E^p$? smooth functions defined around $p$ or... ?
$endgroup$
– user25959
Dec 3 '18 at 1:44
$begingroup$
@user25959 I added a bit of context in the question.
$endgroup$
– NAC
Dec 3 '18 at 1:50