Definition of $C^k( overline{Omega})$
What is the exact definition of $C^k( overline{Omega})$ with $Omega$ open set in $mathbb{R}^n$? The functions in that space have domanin $Omega$ or have domain $overline{Omega}$? Is there a general definition in manifold?
derivatives continuity partial-derivative
asked Nov 22 '18 at 21:45
asvasv
2841211
add a comment |
What is the exact definition of $C^k( overline{Omega})$ with $Omega$ open set in $mathbb{R}^n$? The functions in that space have domanin $Omega$ or have domain $overline{Omega}$? Is there a general definition in manifold?
derivatives continuity partial-derivative
asked Nov 22 '18 at 21:45
asvasv
2841211
The Banach space with norm $|f| = int_Omega (|f(x)|+sum_j |partial_j f(x)|)dx$ has a subspace of continuously differentiable functions on $Omega$. When $Omega$ has a complicated shape there is a smaller subspace which is the space of continuously differentiable functions on some open containing $overline{Omega}$.
– reuns
Nov 22 '18 at 23:57
add a comment |
What is the exact definition of $C^k( overline{Omega})$ with $Omega$ open set in $mathbb{R}^n$? The functions in that space have domanin $Omega$ or have domain $overline{Omega}$? Is there a general definition in manifold?
derivatives continuity partial-derivative
asked Nov 22 '18 at 21:45
asvasv
2841211
What is the exact definition of $C^k( overline{Omega})$ with $Omega$ open set in $mathbb{R}^n$? The functions in that space have domanin $Omega$ or have domain $overline{Omega}$? Is there a general definition in manifold?
derivatives continuity partial-derivative
derivatives continuity partial-derivative
asked Nov 22 '18 at 21:45
asvasv
2841211
asked Nov 22 '18 at 21:45
asvasv
2841211
asked Nov 22 '18 at 21:45
asvasv
2841211
asked Nov 22 '18 at 21:45
asvasv
2841211
asked Nov 22 '18 at 21:45
asvasv
2841211
2841211
The Banach space with norm $|f| = int_Omega (|f(x)|+sum_j |partial_j f(x)|)dx$ has a subspace of continuously differentiable functions on $Omega$. When $Omega$ has a complicated shape there is a smaller subspace which is the space of continuously differentiable functions on some open containing $overline{Omega}$.
– reuns
Nov 22 '18 at 23:57
add a comment |
The Banach space with norm $|f| = int_Omega (|f(x)|+sum_j |partial_j f(x)|)dx$ has a subspace of continuously differentiable functions on $Omega$. When $Omega$ has a complicated shape there is a smaller subspace which is the space of continuously differentiable functions on some open containing $overline{Omega}$.
– reuns
Nov 22 '18 at 23:57
The Banach space with norm $|f| = int_Omega (|f(x)|+sum_j |partial_j f(x)|)dx$ has a subspace of continuously differentiable functions on $Omega$. When $Omega$ has a complicated shape there is a smaller subspace which is the space of continuously differentiable functions on some open containing $overline{Omega}$.
– reuns
Nov 22 '18 at 23:57
The Banach space with norm $|f| = int_Omega (|f(x)|+sum_j |partial_j f(x)|)dx$ has a subspace of continuously differentiable functions on $Omega$. When $Omega$ has a complicated shape there is a smaller subspace which is the space of continuously differentiable functions on some open containing $overline{Omega}$.
– reuns
Nov 22 '18 at 23:57
add a comment |
1 Answer
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The definition I know is
$$
C^k(overline{Omega}) := { vrvert_{overline{Omega}} mid v in C^k(U) quad text{for some open} quad U supset overline{Omega} }.
$$
I don't know about the manifold part.
answered Nov 22 '18 at 23:54
Gonzalo BenavidesGonzalo Benavides
633317
For (differentiable) manifolds it works the same way : for every point $p in Omega$, add to $Omega$ a small neighborhood of $p$ ? For the norm it would need fixing some charts around each $p$, or a metric in the tangent spaces
– reuns
Nov 23 '18 at 0:24
Thank you, have you some textbook reference? Hence the functions in that space have doamin $overline{Omega}$.
– asv
Nov 23 '18 at 11:57
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
The definition I know is
$$
C^k(overline{Omega}) := { vrvert_{overline{Omega}} mid v in C^k(U) quad text{for some open} quad U supset overline{Omega} }.
$$
I don't know about the manifold part.
answered Nov 22 '18 at 23:54
Gonzalo BenavidesGonzalo Benavides
633317
For (differentiable) manifolds it works the same way : for every point $p in Omega$, add to $Omega$ a small neighborhood of $p$ ? For the norm it would need fixing some charts around each $p$, or a metric in the tangent spaces
– reuns
Nov 23 '18 at 0:24
Thank you, have you some textbook reference? Hence the functions in that space have doamin $overline{Omega}$.
– asv
Nov 23 '18 at 11:57
add a comment |
The definition I know is
$$
C^k(overline{Omega}) := { vrvert_{overline{Omega}} mid v in C^k(U) quad text{for some open} quad U supset overline{Omega} }.
$$
I don't know about the manifold part.
answered Nov 22 '18 at 23:54
Gonzalo BenavidesGonzalo Benavides
633317
For (differentiable) manifolds it works the same way : for every point $p in Omega$, add to $Omega$ a small neighborhood of $p$ ? For the norm it would need fixing some charts around each $p$, or a metric in the tangent spaces
– reuns
Nov 23 '18 at 0:24
Thank you, have you some textbook reference? Hence the functions in that space have doamin $overline{Omega}$.
– asv
Nov 23 '18 at 11:57
add a comment |
The definition I know is
$$
C^k(overline{Omega}) := { vrvert_{overline{Omega}} mid v in C^k(U) quad text{for some open} quad U supset overline{Omega} }.
$$
I don't know about the manifold part.
answered Nov 22 '18 at 23:54
Gonzalo BenavidesGonzalo Benavides
633317
The definition I know is
$$
C^k(overline{Omega}) := { vrvert_{overline{Omega}} mid v in C^k(U) quad text{for some open} quad U supset overline{Omega} }.
$$
I don't know about the manifold part.
answered Nov 22 '18 at 23:54
Gonzalo BenavidesGonzalo Benavides
633317
answered Nov 22 '18 at 23:54
Gonzalo BenavidesGonzalo Benavides
633317
answered Nov 22 '18 at 23:54
Gonzalo BenavidesGonzalo Benavides
633317
answered Nov 22 '18 at 23:54
Gonzalo BenavidesGonzalo Benavides
633317
633317
For (differentiable) manifolds it works the same way : for every point $p in Omega$, add to $Omega$ a small neighborhood of $p$ ? For the norm it would need fixing some charts around each $p$, or a metric in the tangent spaces
– reuns
Nov 23 '18 at 0:24
Thank you, have you some textbook reference? Hence the functions in that space have doamin $overline{Omega}$.
– asv
Nov 23 '18 at 11:57
add a comment |
For (differentiable) manifolds it works the same way : for every point $p in Omega$, add to $Omega$ a small neighborhood of $p$ ? For the norm it would need fixing some charts around each $p$, or a metric in the tangent spaces
– reuns
Nov 23 '18 at 0:24
Thank you, have you some textbook reference? Hence the functions in that space have doamin $overline{Omega}$.
– asv
Nov 23 '18 at 11:57
For (differentiable) manifolds it works the same way : for every point $p in Omega$, add to $Omega$ a small neighborhood of $p$ ? For the norm it would need fixing some charts around each $p$, or a metric in the tangent spaces
– reuns
Nov 23 '18 at 0:24
For (differentiable) manifolds it works the same way : for every point $p in Omega$, add to $Omega$ a small neighborhood of $p$ ? For the norm it would need fixing some charts around each $p$, or a metric in the tangent spaces
– reuns
Nov 23 '18 at 0:24
Thank you, have you some textbook reference? Hence the functions in that space have doamin $overline{Omega}$.
– asv
Nov 23 '18 at 11:57
Thank you, have you some textbook reference? Hence the functions in that space have doamin $overline{Omega}$.
– asv
Nov 23 '18 at 11:57
add a comment |
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The Banach space with norm $|f| = int_Omega (|f(x)|+sum_j |partial_j f(x)|)dx$ has a subspace of continuously differentiable functions on $Omega$. When $Omega$ has a complicated shape there is a smaller subspace which is the space of continuously differentiable functions on some open containing $overline{Omega}$.
– reuns
Nov 22 '18 at 23:57