What is the functional consequence of equivalent norms?
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I'm trying to understand the concept of equivalent norms. I've already taken a look at a similar question about this, but do not fully understand.
Two norms are said to be equivalent if:
$ m||v||_b le ||v||_a le M ||v||_b $, for all $v in textbf{V}$ and $ m neq 0 $.
However, I don't think that this literally means that $||v||_1 = ||v||_2$. From reading other posts, my understanding of this is that this means that "two norms that are equivalent induce the same topology".
Can someone give me a higher-level understanding of what this means, suitable for a freshman linear algebra student? What implication does this have for, e.g. applied linear algebra?
linear-algebra
asked Dec 19 '18 at 21:46
GinaGina
183
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add a comment |
$begingroup$
I'm trying to understand the concept of equivalent norms. I've already taken a look at a similar question about this, but do not fully understand.
Two norms are said to be equivalent if:
$ m||v||_b le ||v||_a le M ||v||_b $, for all $v in textbf{V}$ and $ m neq 0 $.
However, I don't think that this literally means that $||v||_1 = ||v||_2$. From reading other posts, my understanding of this is that this means that "two norms that are equivalent induce the same topology".
Can someone give me a higher-level understanding of what this means, suitable for a freshman linear algebra student? What implication does this have for, e.g. applied linear algebra?
linear-algebra
asked Dec 19 '18 at 21:46
GinaGina
183
$endgroup$
$begingroup$
The norms are equivalent in the sense that sequences which converge in one norm converge in the other, or that a subspace with respect to one norm is a subspace with respect to the other.
$endgroup$
– Math1000
Dec 19 '18 at 22:17
$begingroup$
In finite dimensions all norms are equivalent in this sense. But from an applied point of view different norms are very different, e.g. the norm might encode which variables are more or less important. This concept is more useful when dealing with function spaces, since maybe it is easy to show that convergence happens w.r.t. one norm and then you want to show it happens w.r.t. to another (e.g. there are different ways to define Sobolev norms, and they are all equivalent ... But this is not freshman level.)
$endgroup$
– Lorenzo
Dec 20 '18 at 1:16
add a comment |
$begingroup$
I'm trying to understand the concept of equivalent norms. I've already taken a look at a similar question about this, but do not fully understand.
Two norms are said to be equivalent if:
$ m||v||_b le ||v||_a le M ||v||_b $, for all $v in textbf{V}$ and $ m neq 0 $.
However, I don't think that this literally means that $||v||_1 = ||v||_2$. From reading other posts, my understanding of this is that this means that "two norms that are equivalent induce the same topology".
Can someone give me a higher-level understanding of what this means, suitable for a freshman linear algebra student? What implication does this have for, e.g. applied linear algebra?
linear-algebra
asked Dec 19 '18 at 21:46
GinaGina
183
$endgroup$
I'm trying to understand the concept of equivalent norms. I've already taken a look at a similar question about this, but do not fully understand.
Two norms are said to be equivalent if:
$ m||v||_b le ||v||_a le M ||v||_b $, for all $v in textbf{V}$ and $ m neq 0 $.
However, I don't think that this literally means that $||v||_1 = ||v||_2$. From reading other posts, my understanding of this is that this means that "two norms that are equivalent induce the same topology".
Can someone give me a higher-level understanding of what this means, suitable for a freshman linear algebra student? What implication does this have for, e.g. applied linear algebra?
linear-algebra
linear-algebra
asked Dec 19 '18 at 21:46
GinaGina
183
asked Dec 19 '18 at 21:46
GinaGina
183
asked Dec 19 '18 at 21:46
GinaGina
183
asked Dec 19 '18 at 21:46
GinaGina
183
asked Dec 19 '18 at 21:46
GinaGina
183
183
$begingroup$
The norms are equivalent in the sense that sequences which converge in one norm converge in the other, or that a subspace with respect to one norm is a subspace with respect to the other.
$endgroup$
– Math1000
Dec 19 '18 at 22:17
$begingroup$
In finite dimensions all norms are equivalent in this sense. But from an applied point of view different norms are very different, e.g. the norm might encode which variables are more or less important. This concept is more useful when dealing with function spaces, since maybe it is easy to show that convergence happens w.r.t. one norm and then you want to show it happens w.r.t. to another (e.g. there are different ways to define Sobolev norms, and they are all equivalent ... But this is not freshman level.)
$endgroup$
– Lorenzo
Dec 20 '18 at 1:16
add a comment |
$begingroup$
The norms are equivalent in the sense that sequences which converge in one norm converge in the other, or that a subspace with respect to one norm is a subspace with respect to the other.
$endgroup$
– Math1000
Dec 19 '18 at 22:17
$begingroup$
In finite dimensions all norms are equivalent in this sense. But from an applied point of view different norms are very different, e.g. the norm might encode which variables are more or less important. This concept is more useful when dealing with function spaces, since maybe it is easy to show that convergence happens w.r.t. one norm and then you want to show it happens w.r.t. to another (e.g. there are different ways to define Sobolev norms, and they are all equivalent ... But this is not freshman level.)
$endgroup$
– Lorenzo
Dec 20 '18 at 1:16
$begingroup$
The norms are equivalent in the sense that sequences which converge in one norm converge in the other, or that a subspace with respect to one norm is a subspace with respect to the other.
$endgroup$
– Math1000
Dec 19 '18 at 22:17
$begingroup$
The norms are equivalent in the sense that sequences which converge in one norm converge in the other, or that a subspace with respect to one norm is a subspace with respect to the other.
$endgroup$
– Math1000
Dec 19 '18 at 22:17
$begingroup$
In finite dimensions all norms are equivalent in this sense. But from an applied point of view different norms are very different, e.g. the norm might encode which variables are more or less important. This concept is more useful when dealing with function spaces, since maybe it is easy to show that convergence happens w.r.t. one norm and then you want to show it happens w.r.t. to another (e.g. there are different ways to define Sobolev norms, and they are all equivalent ... But this is not freshman level.)
$endgroup$
– Lorenzo
Dec 20 '18 at 1:16
$begingroup$
In finite dimensions all norms are equivalent in this sense. But from an applied point of view different norms are very different, e.g. the norm might encode which variables are more or less important. This concept is more useful when dealing with function spaces, since maybe it is easy to show that convergence happens w.r.t. one norm and then you want to show it happens w.r.t. to another (e.g. there are different ways to define Sobolev norms, and they are all equivalent ... But this is not freshman level.)
$endgroup$
– Lorenzo
Dec 20 '18 at 1:16
add a comment |
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$begingroup$
The norms are equivalent in the sense that sequences which converge in one norm converge in the other, or that a subspace with respect to one norm is a subspace with respect to the other.
$endgroup$
– Math1000
Dec 19 '18 at 22:17
$begingroup$
In finite dimensions all norms are equivalent in this sense. But from an applied point of view different norms are very different, e.g. the norm might encode which variables are more or less important. This concept is more useful when dealing with function spaces, since maybe it is easy to show that convergence happens w.r.t. one norm and then you want to show it happens w.r.t. to another (e.g. there are different ways to define Sobolev norms, and they are all equivalent ... But this is not freshman level.)
$endgroup$
– Lorenzo
Dec 20 '18 at 1:16