Mixing
An alternative way to show that any two norms on a finite dimensional vector space are equivalent. [closed]
$begingroup$
I encountered this different method on page 432 of John Lee's Introduction to Smooth Manifolds. The hint in the book states that first choose an inner product on the vector space, and show that the unit ball in any norm is compact with respect to the topology determined by the inner product. I know how to show the equivalency from this hint. It is easy to show that the new norm can induce a new topology and the unit ball is compact in it. But I have no idea how to show that the unit ball is compact in the original topology. Can anyone help?
compactness normed-spaces inner-product-space
asked Jan 28 at 2:22
user119016user119016
163
$endgroup$
closed as off-topic by Trevor Gunn, Gregory J. Puleo, max_zorn, onurcanbektas, José Carlos Santos Jan 28 at 8:43
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Trevor Gunn, Gregory J. Puleo, max_zorn, onurcanbektas, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
I encountered this different method on page 432 of John Lee's Introduction to Smooth Manifolds. The hint in the book states that first choose an inner product on the vector space, and show that the unit ball in any norm is compact with respect to the topology determined by the inner product. I know how to show the equivalency from this hint. It is easy to show that the new norm can induce a new topology and the unit ball is compact in it. But I have no idea how to show that the unit ball is compact in the original topology. Can anyone help?
compactness normed-spaces inner-product-space
asked Jan 28 at 2:22
user119016user119016
163
$endgroup$
closed as off-topic by Trevor Gunn, Gregory J. Puleo, max_zorn, onurcanbektas, José Carlos Santos Jan 28 at 8:43
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Trevor Gunn, Gregory J. Puleo, max_zorn, onurcanbektas, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
I encountered this different method on page 432 of John Lee's Introduction to Smooth Manifolds. The hint in the book states that first choose an inner product on the vector space, and show that the unit ball in any norm is compact with respect to the topology determined by the inner product. I know how to show the equivalency from this hint. It is easy to show that the new norm can induce a new topology and the unit ball is compact in it. But I have no idea how to show that the unit ball is compact in the original topology. Can anyone help?
compactness normed-spaces inner-product-space
asked Jan 28 at 2:22
user119016user119016
163
$endgroup$
I encountered this different method on page 432 of John Lee's Introduction to Smooth Manifolds. The hint in the book states that first choose an inner product on the vector space, and show that the unit ball in any norm is compact with respect to the topology determined by the inner product. I know how to show the equivalency from this hint. It is easy to show that the new norm can induce a new topology and the unit ball is compact in it. But I have no idea how to show that the unit ball is compact in the original topology. Can anyone help?
compactness normed-spaces inner-product-space
compactness normed-spaces inner-product-space
asked Jan 28 at 2:22
user119016user119016
163
asked Jan 28 at 2:22
user119016user119016
163
edited Jan 28 at 18:07
user119016
asked Jan 28 at 2:22
user119016user119016
163
asked Jan 28 at 2:22
user119016user119016
163
asked Jan 28 at 2:22
user119016user119016
163
163
closed as off-topic by Trevor Gunn, Gregory J. Puleo, max_zorn, onurcanbektas, José Carlos Santos Jan 28 at 8:43
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Trevor Gunn, Gregory J. Puleo, max_zorn, onurcanbektas, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Trevor Gunn, Gregory J. Puleo, max_zorn, onurcanbektas, José Carlos Santos Jan 28 at 8:43
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Trevor Gunn, Gregory J. Puleo, max_zorn, onurcanbektas, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
add a comment |
0
active
oldest
votes
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
